Skip to main content
Log in

The Multidimensional Structure of ‘better than’

  • Original Paper
  • Published:
Axiomathes Aims and scope Submit manuscript


According to the mixed lexicographic/additive account of ‘better than’ and similar aggregative value comparatives like ‘healthier than’, values are multidimensional and different aspects of a value are aggregated into an overall assessment in a lexicographic way, based on an ordering of value aspects. It is argued that this theory can account for an acceptable definition of Chang’s notion of parity and that it also offers a solution to Temkin’s and Rachels’s Spectrum Cases without giving up the transitivity of overall betterness. Formal details and proofs are provided in an “Appendix”.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2

Similar content being viewed by others


  1. This use of the adjective ‘lexicographic’ is customary in decision making. It comes from the way we sort words alphabetically. First, the first letters of two words are compared. If they are the equivalent, then the second letters are compared, and so on.

  2. The theory of value structure is only concerned with abstract structural conditions of value comparisons. It does not address the question whether the underlying value relations are part of the semantics of a natural language expression or somehow pragmatically derived. The theory of value structure provides necessary conditions for a proper semantics or pragmatics of comparative value predicates, but not their complete truth-conditional meaning or a complete specification of their role in speech act content. Investigations of value structure are also normative and not merely descriptive.

  3. See, for example, Gert (2004), Chang (2005), Carlson (2010, 2018), Schoenfield (2014), and Andersson (2017).

  4. See Stojanovic (2015), cf. Smith (2007) about the distinction between taste predicates and predicates of personal taste.

  5. See Stojanovic (2015), McNally and Stojanovic (2017), Weidman Sassoon (2013), and Weidman Sassoon and Fadlon (2017).

  6. There is no agreed-upon terminology, and some authors use these terms in more or less specific senses. We use the more neutral and less common terms ‘dimensions’ and ‘aspects’ to avoid potential misunderstandings.

  7. Intrinsic goodness is no exception because in a realistic scenario items ought not only to be compared according to their intrinsic goodness. There are always extrinsic factors like costs and obligations to consider. For example, even though friendship has intrinsic value, this does not imply that an action or state of affairs is better than another if and only if it promotes more friendship.

  8. To be more precise, these are property instances. For simplicity, the term ‘property’ is used both an abstract concept and particular instances of it throughout this article.

  9. Thanks to Javier Gonzáles de Prado Salas for this example.

  10. This seems to concern Mill’s ‘pleasures of pigs’, but a similar point can be made about too much higher pleasure, though perhaps not from the perspective of Mill (1906) himself. As an anonymous reviewer remarks, probably not all hedonists accept the claim that there is a point at which units of pleasure per duration turn into debauchery. However, it is important not to talk at cross purpose here due to merely conceptual differences. Let us call the above position according to which it can turn into debauchery Hedonism 1. In contrast to this, according to Hedonism 2 too much pleasure is considered pleasure, although it provides disvalue. This is not pleasure in the axiological sense laid out above. According to Hedonism 3, there cannot be enough pleasure and it always provides value. If this pertains to the feeling or experience of pleasure, then we may call it Hedonism 3. Finally, Hedonism 4 defines ‘pleasure’ as ‘whatever provides value.’ This type of hedonism is merely general utilitarianism in disguise. Thus, only Hedonism 3 is in substantive disagreement with Hedonism 1. The matter does not need to be resolved in this article, as both Hedonism 1 and Hedonism 3 are compatible with MLTB. Whoever is inclined towards Hedonism 3 should ignore the above type 1 hedonist examples.

  11. Maybe it is not, if we tell the grieving widow about the lollipops, but let us put such complications aside for the sake of the argument. Of course, lollipops could be replaced with any kind of small and relatively unimportant benefits in these examples.

  12. This is one of many possible solutions to the Repugnant Conclusion and not the one recommended or endorsed here. There are many more problems with sum utilitarianism, but this discussion belongs elsewhere.

  13. Cf. Chang (2012).

  14. See Luce (1956), Tversky (1969), Fishburn (1970a), Vincke and Pirlot (1997), Bouyssou and Vincke (2009). In contrast to semiorders, interval relations can also represent nested intervals. We don’t need this property in MLTB.

  15. In Luce’s paper semiorders are interpreted as an agent’s indifference, which is weaker than believing that two items are equally good. As of the time of this writing, the author is not aware of any decisive normative arguments for or against taking semiorders as base relations for equal goodness. Notice, however, that the above justification is epistemic and not merely psychological, since the presence of measurement inaccuracies within a given value dimension can make the use of semiorder representations inevitable.

  16. Cf. Carlson (2018, p. 525). Objections against the hedonic underpinnings of the example or counter-arguments based on the scenarios’ realism should be ignored. Spectrum Cases can be reformulated using other dimensions such as welfare levels and happiness, and any levels and durations can be chosen. All that is needed is a duration that is much longer than that of the previous item, combined with a level that is just a little bit lower than the previous item.

  17. This would open an avenue to interesting robustness analyses of uses of ‘better than.’ If Aspect 1 is lexicographically preferred to Aspect 2, but it is unclear whether an item belongs more to 1 or more to 2, then maybe we should withhold judgment and aggregation must stop. This is left for another occasion.

  18. A fourth alternative threatens rationality more than any of the other proposals. It is based on the idea that we do not have to choose maximally consistent sets of potentially conflicting rationality postulates, but instead somehow manage to maintain them in equilibrium, picking the ‘fitting ones’ at specific occasions. This ‘normative dialethism’ seems to be based on a confusion between a normative conception of rationality and the conditions for acting rationally, which are a psychological matter. However, the former is logically prior to the latter.

  19. MLTB is unipolar, meaning that a is worse than b if and only if b is better than a. So ‘worse than’ is not really needed. This assumption is commonly made, as it is hard to axiologically justify bipolar theory in which this equivalence does not hold.

  20. See Chang (2002, pp. 667–668).

  21. Chang (2005)’s formal argument against Gert (2004) may work against his peculiar use of interval orders, but is generally rather weak. If a is on a par with a, making \(a^+\) better than a implies that \(a^+\)’s lower boundary in the interval representation is larger than a’s higher boundary, and so \(a^+\) cannot be on a par with a according to SIA. However, the idea behind an interval representation is that \(a^+\) could be improved by making it overlap without being strictly better than a, i.e., by having a lower bound that is higher than a’s lower bound but lower than a’s upper bound and an upper bound that is higher than a’s upper bound. Distinguishing between ‘improving’ and ‘making strictly better than’ solves the problem. However, in this view parity remains unidimensional and thus does not do justice to the fact that in Chang’s own examples items are compared in aspects that are related, but also different from each other in subtle ways, such as the creativity of a painter versus the creativity of a musician.

  22. See Hansson (2002, p.  337), Hansson (2001, pp. 24–25).

  23. See, for example, Broome (2004, p. 51); cf. Carlson (2018, pp. 525–526).

  24. See Hansson (2018, pp. 567–571).

  25. Extensions of MLTB to deal with cardinal utilities are discussed in Appendix “Other Kind of Value Structures”.

  26. Work on this article was conducted with funding by the Portuguese Foundation for Science and Technology (FCT) and the New University of Lisbon under grant PTDC/MHC-FIL/0521/2014 and individual grant DL 57/2016/CP1453/CT0002. Many thanks to Pedro Abreu, Per Algander, Erik Carlson, Javier Gonzáles de Prado Salas, António Zilhão, members of the Higher Seminar of Practical Philosophy at Uppsala University, the members of the FCT project “Values in Argumentative Discourse”, participants of the IFILNOVA Value Seminar, the ArgLab Colloqium, and the Reading Group in Ethics and Political Philosophy at the New University of Lisbon, the members of the CFCUL Reasoning Group of the University of Lisbon, as well as several anonymous reviewers for helpful discussion, suggestions, and comments.

  27. See Bouyssou and Vincke (2009, p. 52).

  28. See for example Bouyssou et al. (2009, pp. 796–798).

  29. The introduction of weights may seem questionable from a measurement-theoretic perspective, but is unavoidable. Aspects of ‘better than’ comparisons can be more or less important. Even though the weights may be set to 1 and thus ignored, this would be axiologically implausible as a general solution. Bear in the mind that we are not in the business of empirical measurement but are developing a normative theory of ‘better than’ at this stage.

  30. Cf. Eisenführ et al. (2010, pp. 130–134), see also Keeney and Raiffa (1976, pp. 108–112). The above formulation of preferential independence is based on conjoint independence in Fishburn and Wakker (1995).

  31. The method would not work for uncountable domains. As Debreu (1954, p. 105, fn. 1) shows, lexicographic preferences violate the condition of order separability needed to guarantee the existence of a utility function for uncountable domains.

  32. See Bouyssou et al. (2009, p. 791), cf. Arrow (1951), Bouyssou (2003).

  33. See Conitzer and Davenport (2006).

  34. See Choquet (1954), Sugeno (1974, (1977); cf. Marichal (2009, pp. 700–709).

  35. See Roubens and Vincke (1985). As Vincke and Pirlot (1997) lay out, the threshold could also be a function of a. What distinguishes semiorders from interval orders is that the former cannot represent nested intervals.

  36. Notice also that u(.) is not normalized, which further complicates attempts to introduce such a threshold.

  37. See (Roberts 1979, p. 64).


  • Abdellaoui M, Gonzales C (2009) Multiattribute utility theory. In: Bouyssou D, Dubois D, Prade H, Pirlot M (eds) Decision-making process. Wiley, London/Hoboken, pp 579–616

    Chapter  Google Scholar 

  • Andersson H (2017) How it all relates: exploring the space of value comparisons. Doctoral dissertation, Lund University, Sweden, Lund

  • Arrow KJ (1951) Social choice and individual values. Yale University Press, New Haven

    Google Scholar 

  • Bogart KP (1973) Preference structures I: distances between transitive preference relations. J Math Sociol 3:49–67

    Article  Google Scholar 

  • Bouyssou D (2009) Outranking methods. In: Floudas CA, Pardalos PM (eds) Encyclopedia of optimization. Springer, New York, pp 2887–2893

    Chapter  Google Scholar 

  • Bouyssou D, Marchant T, Perny P (2009) Social choice theory and multicriteria decision aiding. In: Bouyssou D, Dubois D, Prade H, Pirlot M (eds) Decision-making process. Wiley, London/Hoboken, pp 779–810

    Chapter  Google Scholar 

  • Bouyssou D, Vincke P (2009) Binary relations and preference modeling. In: Buouyssou D, Dubois D, Prade H, Pirlot M (eds) Decision-making process, Chapter 2. Wiley, London, pp 49–84

    Chapter  Google Scholar 

  • Bouyssou D, Pirlot M (2003) Ordinal aggregation and strict preferences for multi-attributed alternatives. Internal Report Cahier du LAMSADE 212, Université Paris Dauphine, Paris, France

  • Broome J (2004) Weighing Lives. Oxford University Press, Oxford

    Book  Google Scholar 

  • Carlson E (1996) Cyclical preferences and rational choice. Theoria 62(1–2):144–160

    Google Scholar 

  • Carlson E (2010) Parity demystified. Theoria 76:119–128

    Article  Google Scholar 

  • Carlson E (2018) Value theory (axiology). In: Hansson SO, Hendricks VF (eds) Introduction to formal philosophy. Springer, Berlin, pp 523–534

    Chapter  Google Scholar 

  • Chang R (ed) (1997) Incommensurability, incomparability, and practical reason. Harvard University Press, Cambridge

    Google Scholar 

  • Chang R (2002) The possibility of parity. Ethics 112:669–688

    Article  Google Scholar 

  • Chang R (2005) Parity, interval value, and choice. Ethics 115:331–350

    Article  Google Scholar 

  • Chang R (2012) Are hard choices cases of incomparability? Philos Issues 22(1):106–126

    Article  Google Scholar 

  • Choquet G (1954) Theory of capacities. In: Annals of Institute Fourier, Grenoble. Nundam, vol 5, pp 131–295

  • Conitzer V, Davenport A (2006) Improved bounds for computing Kemeny rankings. In: Cohn A (ed) AAAI’06 Proceedings of the 21st national conference on artificial intelligence—volume 1. American Association for Artificial Intelligence, pp 620–626

  • Debreu G (1954) Representation of preference ordering by a numerical function. In: Thrall R, Coombs C, Davies R (eds) Decision processes. Wiley, New York, pp 159–175

    Google Scholar 

  • Eisenführ F, Weber M, Langer T (2010) Rational decision making. Springer, Berlin

    Book  Google Scholar 

  • Fishburn PC (1970a) Intransitive indifference in preference theory: a survey. Oper Res 18(2):207–228

    Article  Google Scholar 

  • Fishburn PC (1970b) Utility theory for decision making. Wiley, New York

    Book  Google Scholar 

  • Fishburn PC (1972) Axioms for lexicographic preferences. Technical Report, Office of Naval Research, Washington, DC. ERIC Number: ED072807, Accession Number: AD0752489, Pennsylvania State University, College of Business Administration, National Technical Information Service, Springfield, VA

  • Fishburn PC (1974) Lexicographic orders, utilities and decision rules: a survey. Manag Sci 20(11):1442–1471

    Article  Google Scholar 

  • Fishburn PC (1975) Axioms for lexicographic preferences. Rev Econ Stud 42(3):415–419

    Article  Google Scholar 

  • Fishburn PC, Wakker PP (1995) The invention of the independence condition for preferences. Manag Sci 41(7):1130–1144

    Article  Google Scholar 

  • Gert J (2004) Value and parity. Ethics 114:492–510

    Article  Google Scholar 

  • Handfield T (2016) Essentially comparative value does not threaten transitivity. Thought 5(1):3–12

    Article  Google Scholar 

  • Hansson SO (2001) The structure of values and norms. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Hansson SO (2002) Preference logic. In: Gabbay DM, Guenthner F (eds) Handbook of philosophical logic, vol 4, 2nd edn. Kluwer, Dordrecht, pp 319–393

    Google Scholar 

  • Hansson SO (2018) Money-pumps. In: Hansson SO, Hendricks VF (eds) Introduction to formal philosophy. Springer, Cham, pp 567–576

    Chapter  Google Scholar 

  • Jacquet-Lagréze E (1975) How we can use the notion of semiorders to build outranking relations in multi-criteria decision making. In: Wendt D, Vlek C (eds) Utility, probability, and human decision making. D. Reidel, Dordrecht, pp 87–112

    Chapter  Google Scholar 

  • Keeney RL, Raiffa H (1976) Decisions with multiple objectives: preferences and value tradeoffs. Wiley, New York

    Google Scholar 

  • Kemeny J (1959) Mathematics without numbers. Daedalus 88(4):577–591

    Google Scholar 

  • Kemeny John, Snell JL (1972) Mathematical models in the social sciences. MIT Press, Cambridge

    Google Scholar 

  • Klocksiem J (2016) How to accept the transitivity of better than. Philos Stud 173:1309–1334

    Article  Google Scholar 

  • Levi I (1986) Hard choices. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Luce DR (1956) Semiorders and a theory of utility discrimination. Econometrica 24(2):178–191

    Article  Google Scholar 

  • Mandler M (2005) Incomplete preferences and rational intransitivity of choice. Games Econ Behav 50:255–77

    Article  Google Scholar 

  • Marichal J-L (2009) Aggregation functions for decision making. In: Bouyssou D, Dubois D, Pirlot M, Prade H (eds) Decision-making process: concepts and methods. Wiley, London, pp 673–721

    Chapter  Google Scholar 

  • McNally L, Stojanovic I (2017) Aesthetic adjectives. In: Young J (ed) The semantics of aesthetic judgment. Oxford University Press, Oxford, pp 17–37

    Chapter  Google Scholar 

  • Mill JS (1906) Utilitarianism. Chicago University Press, Chicago

    Google Scholar 

  • Parfit D (1984) Reasons and persons. Oxford University Press, Oxford

    Google Scholar 

  • Rabinowicz W (2008) Value relations. Theoria 74:18–49

    Article  Google Scholar 

  • Rabinowicz W (2010) Value relations: old wine in new barrels. In: Reboul A (ed) Philosophical papers dedicated to Kevin Mulligan. Université de Genève, Genève

    Google Scholar 

  • Rabinowicz W (2012) Value relations revisited. Econ Philos 28(2):133–164

    Article  Google Scholar 

  • Rabinowicz W (2016) Aggregation of value judgments differs from aggregation of preferences. In: Kuźniar A, Odrow aż-Sypniewska J (eds) Uncovering facts and values: studies in contemporary epistemology and political philosophy, Poznań studies in the philosophy of the sciences and the humanities (107). Brill/Rodopi, Leiden

    Google Scholar 

  • Rachels S (1998) Counterexamples to the transitivity of better than. Austral J Philos 76(1):71–83

    Article  Google Scholar 

  • Rachels S (2001) A set of solutions to parfit’s problems. Nous 35(2):214–238

    Article  Google Scholar 

  • Rawls J (1971) A theory of justice. Belknap Press, Cambridge

    Book  Google Scholar 

  • Roberts FS (1979) Measurement theory. Adison Wesley, Reading

    Google Scholar 

  • Roubens M, Vincke P (1985) Preference modelling. Springer, Heidelberg

    Book  Google Scholar 

  • Schoenfield M (2014) Decision making in the face of parity. Philos Perspect 28(1):263–277

    Article  Google Scholar 

  • Schumm GF (1987) Transitivity, preference and indifference. Philos Stud 52:435–437

    Article  Google Scholar 

  • Smith BC (2007) The objectivity of taste and tasting. In: Smith BC (ed) Questions of taste: the philosophy of wine. Oxford University Press, Oxford, pp 41–77

    Google Scholar 

  • Stojanovic I (2015) Evaluative adjectives and evaluative uses of ordinary adjectives. In: Bekki D, McCready E (eds) Proceedings of LENLS12: language engineering and natural language semantics. The Japan Society for Artificial Intelligence

  • Sugeno M (1974) Theory of fuzzy integrals and its applications. PhD thesis, Tokyo Institute of Technology, Tokyo, Tokyo

  • Sugeno M (1977) Fuzzy measures and fuzzy integrals: a survey. In: Fuzzy automata and decision process. North Holland, New York

  • Temkin LS (1987) Intransitivity and the mere addition paradox. Philos Public Aff 16(2):138–187

    Google Scholar 

  • Temkin LS (2012) Rethinking the good: moral ideals and the nature of practical reasoning. Oxford University Press, New York

    Book  Google Scholar 

  • Tversky A (1969) Intransitivity of preferences. Psychol Rev 76(1):31–48

    Article  Google Scholar 

  • Vincke P, Pirlot M (1997) Semiorders: properties, representations, applications. Springer, Dordrecht

    Google Scholar 

  • von Wright GH (1963) The varieties of goodness. Routledge & K. Paul, London

    Google Scholar 

  • Voorhoeve A (2013) Vaulting intuitions: Temkin’s critique of transitivity. Econ Philos 29(3):409–423

    Article  Google Scholar 

  • Weidman Sassoon G (2013) A typology of multidimensional adjectives. J Semant 30:335–380

    Article  Google Scholar 

  • Weidman Sassoon G, Fadlon J (2017) The role of dimensions in classification under predicates predicts their status in degree constructions. Glossa 2(1)(42):1–40

    Google Scholar 

  • Williamson T (1994) Vagueness. Routledge, London

    Google Scholar 

  • Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353

    Article  Google Scholar 

Download references


This work was supported by the Portuguese Fundação para a Ciência e a Tecnologia (Grant No. DL 57/2016/CP1453/CT0002).

Author information

Authors and Affiliations


Corresponding author

Correspondence to Erich H. Rast.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


Appendix 1: Auxiliary Definitions and Select Theorems

It is assumed that the domain of items under consideration is finite and that a finite number of aspect relations partially orders these. It is not assumed that the description of the respective value dimension is natural or simple—the description of a dimension may be complex. The following properties of binary relations are used:

Definition 1

(Properties of Relations) A binary relation \(R\subseteq D\times D\) is

  • complete iff. \(aRb\vee bRa\)

  • reflexive iff. aRa

  • symmetric iff. \(aRb\Rightarrow bRa\)

  • transitive iff. \((aRb\, \& \, bRc)\Rightarrow aRc\)

  • semitransitive iff. \((aRb\, \& \, bRc)\Rightarrow (aRd\vee dRc)\)

  • Ferrers iff. \((aRb\, \& \, cRd)\Rightarrow (aRd\vee cRb)\)

for any \(a, b, c, d\in D\).Footnote 27

The following select theorems show that multidimensional parity has some of the properties laid out by Chang.

Theorem 1

(Non-Transitivity of Rough Equality) Semiorder-based equality is not transitive.(Luce 1956)


By example. Let R be a semiorder and define \(aIb:=aRb\, \& \, bRa\). Define aRb, bRa, bRc, cRb, cRd, and dRc. By definition we have aIb and bIc but not aIc, since there is no link from a to c. (i) R is semitransitive, since for aRb and bRc it is also the case that cRd, and for cRb and bRa it also holds that cRd, and for dRc and cRb it also holds that bRc. (ii) R is Ferrers, since for aRb and cRd it also holds that cRb, and for dRc and bRa it also holds that aRc. Hence, R is a semiorder with a weak part that is not transitive. \(\square\)

Theorem 2

Parity is not transitive.


It suffices to show that aspect parity is not transitive. Consider the case when \(a,c\in S_1\), \(b\in S_2\), \(b\notin S_1\), abc are in both \(D_1\) and \(D_2\), and add an arbitrary element to \(S_2\) to fulfill the conditions for an LES. Assume that \(|T(a, 1)-T(b, 2)|\le \delta\) and \(|T(b, 2)-T(c, 1)|\le \delta\). Then a is on a par with b and b is on a par with c, but a is not on a par with c. \(\square\)

Note that a and c need not even be roughly equal in this case, since the threshold k is independent of the underlying interval threshold for the internal equality relation \(I_1\) of the first aspect.

Theorem 3

Aspect parity is symmetric.


By definition of aspect parity and because \(|x-y|=|y-x|\): \(|T(a, i)-T(b, j)|=|T(b, j)-T(a, i)|\) for any \(a, b\in {\mathcal {C}}\) and \(i, j\in A\). \(\square\)

Theorem 4

Parity is symmetric.


This follows directly from the definition of parity and the symmetry of aspect parity.\(\square\)

Theorem 5

Three items can be on a par with respect to three aspects.


By example. Let \(a\in S_1\), \(N_2(a), N_3(a)\), \(b\in S_2\), \(N_1(b)\), \(N_3(b)\), and \(c\in S_3\), \(N_1(c)\), \(N_2(c)\), and choose the respective top distances such that \(|T(a, 1)-T(b, 2)|\le \delta\), \(|T(a, 1)-T(c, 3)|\le \delta\), and \(|T(b, 2)-T(c, 3)|\le \delta\). \(\square\)

The next theorem shows a combinatorial limit of this model of parity.

Theorem 6

If n distinct items are aspectually on a par with each other, then the underlying value structure has at least n distinct aspects.


We assume that (i) there are n items that are on a par with each other but (ii) the value structure has only \(k<n\) aspects, and derive a contradiction.

Case 1: Suppose \(n=1\). Then k is 0 and no items can be on a par. Case 2: Suppose \(n=2\). Then k is 1 or lower and by definition of parity the two items cannot be on a par either. Case 3: Suppose \(n>2\). Without loss of generality we assume the maximal number of aspects, i.e., that \(k=n-1\). Name the items to be on a par \(x_1, x_2, \ldots , x_n\) and the aspects \(1, 2, \ldots , k\). Let us write \(-i\) for all indices j such that \(j\ne i\) and \(1\le i, j\le k\). With a bit of abuse of notation, we can map any item \(x_i\) to the k aspects by setting \(|T(x_i, S_i)-T(x_{-i}, S_{-i})|<\delta\) and \(x_i\in S_i\) while at the same time \(x_{-i}\in D_i\) and \(N_i(x_{-i})\), such that \(x_i\) is on a par with all \(x_{-i}\). By assumption, \(x_{k+1}\) is also on a par with all of these items. However, by the definition of aspect parity \(x_{k+1}\) then must be in some \(S_j\) and by the Pigeonhole Principle there is already an item \(x_j\in S_j\) such that \(j\ne k+1\). Hence, by definition of aspect parity \(x_{k+1}\) and \(x_j\) cannot be on a par, contradicting the assumption. \(\square\)

Theorem 6 illustrates that parity is a fairly demanding notion. However, together with Theorem 2 the next theorem establishes that this kind of parity matches Chang’s intuitions about the relation in Chang (2002).

Theorem 7

(Compatibility with SIA) Aspect parity may be preserved under small improvements. If ab are on a par with respect to aspects 1, 2, and a is improved within aspect 1 to \(a^+\), then \(a^+\) and b may be on a par with respect to aspects 1, 2.


Suppose \(N_2(a)\), \(a\in S_1\), \(N_1(b)\), \(b\in S_2\), and \(|T(a, 1)-T(b,2)|\le \delta\). Let \(a^+P_1a\). Then it follows from this and the definition of top distance that \(|T(a^+, 1)-T(b, 2)|\le \delta\) may be fulfilled, too. For example, the condition is fulfilled if \(T(b,2)\le T(a,1)\), because it follows from the definition of top distance that in this case \(T(a^+, 1)<T(a, 1)\). \(\square\)

Note that if \(\delta \ge 1\), then an improvement \(a\mapsto a^+\) may turn out to be too large to preserve ij-aspect parity with b only if \(T(b, j)> T(a, i)\), since the top distance of an item improved in an aspect i will always be lower than that of the original item and the minimum distance between two items is 1.

Appendix 2: Value Aggregation

In this “Appendix” , as a proof of concept a value aggregation method is laid out that takes into account lexicographic hierarchies of aspects while at the same time allowing for more traditional aggregation within each lexicographic level.

1.1 Aggregation for Ordinal Preorder Value Structures

We begin by assuming the completeness of ‘\(\succeq\)’, the preorder relation over the aspects A of a lexicographic value structure, and look at the case when the relation is incomplete later. The strict part of this relation is written as ‘\(\succ\)’ and the symmetric part as ‘\(\sim\).’ An aspect level function \(\ell : A\rightarrow \mathbb {N}\) for the aspects is defined for ‘\(\succeq\)’ like in the definition of function L. The lexicographic equivalence class of an aspect level is then defined as \(Eq(x):=\{i\in \mathbb {N}\mid \ell (i)=x\}.\)

To canonically construct a utility function \(u_i: D\rightarrow \mathbb {R}\) for an aspect, a similar method is used, but this time the ranking function needs to be averaging the rank at each level in order to allow us to normalize the function to the number of comparable items. This is important for making canonical ordinal utilities comparable with each other. For simplicity, only the case when the domain is finite is considered in what follows.

With respect to aspect \(i\in A\), \(x\in S_i\), \(Eq_i(x):=\{y\in S_i\mid L_i(x)=L_i(y)\}\) is called the apathy class of item x. \(E_i(x):=\{y\in Eq_i(z)\mid \text {for any }z\text { such that }L(z)=x\}\) is the apathy class at level x under aspect i, and a starting index function at a level is defined recursively as follows:

  1. 1.

    \({\mathcal {O}}_i(1)=1\)

  2. 2.

    \({\mathcal {O}}_i(x+1)={\mathcal {O}}_i(x)+|E_i(x+1)|\).

Based on these auxiliary definitions, we can define the averaging Borda rank for levels and items. The Borda Rank with averaging ties, abbreviated as ‘averaging Borda rank or just ‘rank’ in what follows, is defined for a given aspect k and level x as

$$\begin{aligned} B_k(x)=\frac{1}{|E_k(x)|}{{\sum }}_{i={\mathcal {O}}_k(x)}^{{\mathcal {O}}_k(x)+|E_k(x)|-1} i=\frac{1}{2}(|E_k(x)|+2{\mathcal {O}}_k(x)-1). \end{aligned}$$

A corresponding function for items \(x\in S_k\) is defined based on an item’s level as \(v_k(x)=B_k(L_k(x))\).

In the literature on Social Choice, the Borda rank is often defined simpler and the other way around, starting from 1 (best) to n (worst) item such that the more preferred item has a lower score than the less preferred item.Footnote 28 However, the above formulation reveals an important clue for normalization. Function \(v_i(.)\) represents the ordinal value of an item relative to other items with respect to the aspect i. Note that the above analytic formula for the Borda rank is an instance of the arithmetic series

$$\begin{aligned} S(a_i)=\frac{1}{k+1}{{\sum }}_{i=m}^{m+k}i=\frac{1}{2}(2m+k), \end{aligned}$$

which is a generalization of the famous Euler solution for summing the integers from \(1, 2, \ldots , n\):

$$\begin{aligned} \sum _{i=1}^n i=\frac{1}{2}n(n+1). \end{aligned}$$

This fact allows us to create normalized canonical utility functions that are independent of the size of the domain \(S_i\) of comparable items of an aspect.

Definition 2

(Canonical Ordinal Utility Function) For each aspect i, ordinal utility is defined as the averaging Borda rank for comparable items \(x\in S_i\):

$$\begin{aligned} u_i(x):=\frac{2v_i(x)}{|S_i|(|S_i|+1)}. \end{aligned}$$

The following theorem establishes that normalizing in this way is adequate.

Theorem 8

(Analytic Sum of Averaging Borda Rank) The following equality holds for any aspect k and \(n=|S_k|\):

$$\begin{aligned} {\sum }_{x\in S_k} v_k(x)={\sum }_{i=1}^n i=1+2+\cdots +n-1+n=\frac{1}{2}n(n+1) \end{aligned}$$


From the definition of an item’s level with respect to an aspect we know that every item resides at one and only one level. Therefore, we can proceed with definition \(B_i(x)\) for Borda ranking and for brevity leave out any references to aspects in what follows. We simplify (1) by setting \(k=|E(x)|-1\) and \(m={\mathcal {O}}(x)\), obtaining the formula for the arithmetic series (2) as the score for an apathy class at some level based on the comparisons

$$\begin{aligned} \ldots a_mIa_{m+1}Ia_{m+2}I\ldots Ia_{m+k}\ldots \end{aligned}$$

When \(k=0\) at each level, i.e., when the underlying ordering is strict, we obtain an application of (3), since then at each level we trivially get \(\sum _{i=m}^{m+0} i=m\) as the rank of the item at that level. What is left to show is that the sum of the ranks of the items in (6) is identical to the sum of the ranks in the strict ordering

$$\begin{aligned} \ldots a_{m+k}Pa_{m+k-1}P\ldots Pa_{m+1}Pa_m\ldots \end{aligned}$$

But the sum of the ranks in (7) is just an instance of the general sequence (2) from item \(a_m\) to item \(a_{m+k}\), and if we substitute back k and m the last item in this sequence is \(a_{{\mathcal {O}}(x)+|E(x)|-1}\) like the last item in (1), and the first item is \(a_{{\mathcal {O}}(x)}\) like in (1). Hence, the sums for (6) and (7) are instances of the same series (2) with the same start and end, and thus identical. So if we sum over all items in (1) the result is the same as summing over a corresponding strict ordering whose sum is given by (3). \(\square\)

In (4) of Definition 2 we divide the Borda rank \(v_i(x)\) of item x with respect to aspect i by the analytic maximum of the sum of the Borda ranks of items in i given by formula (3), and the above theorem shows that the averaging Borda rank has the same maximum. Thus, canonical utilities become comparable in the sense that they all reside within the interval [0, 1] and the size of the sets of comparable items does not introduce some inadequate implicit weight.

We proceed to define a lexicographic aggregation function based on these ordinal utilities that ensures that the level of the aspects given by ordering ‘\(\succeq\)’ is respected, but aggregates ordinal utilities within the same aspect level in a traditional way by computing a weighted sum.Footnote 29 The result is a mixture of additive and lexicographic aggregation.

Let each aspect at aspect level k have some weight \(w_i\) such that \(w_i>0\) and the sum of all \(w_i\) at k is 1. The utility of an item x at aspect level k given by ‘\(\succeq\)’ is

$$\begin{aligned} u^k(x) := \sum _{i\in Eq(k)} w_iu_i(x) \end{aligned}$$

The maximum utility at an aspect level k is defined as:

$$\begin{aligned} M(k):=\max _x u^k(x) \end{aligned}$$

This requires all relations \(R_i\in Eq(k)\) to be preferentially independent in the following sense. At any aspect level k and for any items \(a, b, a', b'\), consider any case in which relations \(R_1,\ldots , R_n\in Eq(k)\) can be partitioned into two sets \({\mathcal {R}}\) and \({\mathcal {I}}\) such that for every \(R_x\in {\mathcal {I}}\) we have \(aI_xb\) and \(a'I_xb'\) and for every \(R_y\in {\mathcal {R}}\) we have \(aI_ya'\) and \(bI_yb'\). Preferential independence holds at k if in any such case \(u^k(a)\ge u^k(b)\) if and only if \(u^k(a')\ge u^k(b')\). Only if this condition is fulfilled, can Definition 8 guarantee that the relations are faithfully aggregated at level k.Footnote 30

Using concepts introduced so far, value aggregation at the highest applicable level can take into account the maxima of all previous levels. For all \(x\in D\) such that there is a highest aspect level k at which some \(u_i(x)\) is defined, the aggregate utility \(u:D\rightarrow \mathbb {R}\) is

$$\begin{aligned} u(x):=u^k(x)+M(k-1)+M(k-2)+\cdots +M(1). \end{aligned}$$

If there is no \(u_i(.)\) defined at x for any \(i\in A\), then u(x) is undefined at point x.

This definition does not rely on hyperreal numbers and nonstandard analysis like other methods such as Fishburn (1972, 1974), making it somewhat easier lay out and use but comes at the price of loosing mathematical insight and generality.Footnote 31 The construction of u(.) guarantees for finite domains that if \(S_i\succ S_j\), \(a\in S_i\) and \(b\in S_j\), then \(u(a)>u(b)\). So, the evaluation is lexicographic, as the following theorem establishes.

Theorem 9

(Mixed Aggregation is Lexicographic) If \(i\succ j\) for some aspects ij in a lexicographic value structure, then \(u(x)>u(y)\) for any \(x\in S_i, y\in S_j\) regardless of the evaluation by other aspects of x and y.


Assume the antecedent \(i\succ j\). This implies \(\ell (i)>\ell (j)\). Suppose, without loss of generality, that \(\ell (i)=\ell (j)+1\). Then (a) \(u(y)=u^{i-1}(y)+M\big (\ell (i-2)\big )+\cdots +M\big (1\big )\), and (b) \(u(x)=u^{i}(x)+M\big (\ell (i-1)\big )+M\big (\ell (i-2)\big )+\cdots +M\big (1\big )\). Moreover, it follows from Definition 1 that \(u_i(x)\) is positive for any i and x, and in turn by 10 that \(u^k(x)\) is positive for any x and aspect level k it is defined. From this positivity in combination with (a) and (b) it follows that \(u(x)>u(y)\) even if \(u^{j}(x)=M(\ell (j))\), because u(x) is by definition larger than the sum of the maximum utilities at levels \(\ell (j), \ell (j-1), \ldots , 1\). \(\square\)

Theorem 10

(Transitivity of Overall Betterness) The aggregate ‘better than’ relation R defined as aRb iff. \(u(a)\ge u(b)\) is reflexive and transitive.


Follows directly form the fact that u(.) is a function into the real numbers and ‘\(\ge\)’ is reflexive and transitive.\(\square\)

This result confirms Klocksiem (2016)’s thesis that lexicographic ‘better than’ with absolute thresholds can maintain transitivity while accepting the intuitions suggested by Spectrum Cases, expanding his thesis to multidimensional betterness. It is further worth noting that it is well-known from Social Choice that the Borda method violates the Independence of Irrelevant Alternatives axiom of Arrow’s Theorem and therefore does not lead to negative consequences like dictatorial or oligarchic preferences in the above application.Footnote 32

However, it is worth noting that the proposed aggregation is not the only possible method and, generally speaking, the problem of how to normatively justify a particular method of value aggregation at a given lexicographic aspect level remains open. As an alternative to the method laid out above, minimization of a distance measure may also be used. Kemeny (1959) and Kemeny (1972) proposed a modified inversion measure since then sometimes called ‘Kemeny rank’, which Bogart (1973) generalizes to incomplete strict orders and which is used by Rabinowicz (2016) for some form of social value aggregation. Kemeny distance measure is computationally expensive, though,Footnote 33 whereas other measures like Kendall’s \(\tau\) would be ad hoc for value aggregation without further justification. In a broader setting, one would also have to take into account the Choquet integral for the non-additive aggregation of cardinal utilities and the Sugeno integral for the non-additive aggregation of ordinal utilities.Footnote 34

1.2 Other Kind of Value Structures

By the theorems of Debreu (1954) the existence of an ordinal utility function is only guaranteed for a complete preorder relation. In a semiorder value structure the utility representation changes. Since base relations \(R_i\) may also be incomplete in the current setting, the following conditionals hold for some constant k:Footnote 35

$$\begin{aligned} aP_ib&\Rightarrow u(a)-u(b)\ge k \end{aligned}$$
$$\begin{aligned} aI_ib&\Rightarrow u(a)-u(b)<k \end{aligned}$$

Complete relations turn these representation conditions into biconditionals. The mixed aggregation is unaffected by this change and the Borda Count method is perfectly reasonable as an aggregation method for ordinal preferences of this kind. Since incomparable items are not taken into account, no changes to the definition of lexicographic aggregation are needed. Attempting to define a semiorder from the result u(.) instead of a preorder like in Theorem 10 seems questionable, though; in any case, the threshold would have to be motivated independently from the thresholds of the base utilities due to the way (10) works.Footnote 36

Next, we take a look at the case when ‘\(\succeq\)’ is incomplete. This case is problematic. Consider the set of sets of \(\succeq\)-comparable aspects:

$$\begin{aligned} A^p:=\{X\subseteq A\mid i,j\in X\Leftrightarrow i\succeq j\text { or }j\succeq i\} \end{aligned}$$

How should the definition of lexicographic aggregation be changed to deal with the case when \(A^p\) contains more than one set? For instance, suppose that \(i\succ j\) and \(k\succ j\) but i and k are not lexicographically ordered. Although it would be trivial to change the definition such that different members of this set are aggregated separately and then the results are aggregated somehow, this would not be faithful to the intended interpretation of the case when two aspects are not comparable by ‘\(\succeq\).’ What we should say is that the utility of two items ab can only be compared if the there are no aspects ij in the value structure such that \(i\in X\), \(j\in Y\), \(X\ne Y\), and \(X, Y\in A^p\), and \(u_i(a)\) and \(u_j(b)\) are defined. Still, when \(|A^p|>1\), there are such incomparable items and the value structure is deficient. To avoid such problems, it seems best to consider only complete lexicographic value structures in which ‘\(\succeq\)’ is complete, and give up aspect and value noncomparability. Since two values can still be incommensurable if all their aspects are incommensurable and we can also distinguish incommensurability from mutual exclusivity, not much is lost in expressivity.

Finally, the more pressing issue of cardinal utilities shall be addressed. There are two cases to consider. In the first case some, but not all aspects are based on a cardinal dimension and in the second case all aspects are based on a cardinal dimension. Starting with the second case, every utility \(u_i(.)\) then must either represent items in the sense of (11) and (12) in a semiorder value structure, or according to the standard condition modified for incomplete base relations in the following sense:

$$\begin{aligned} aR_ib&\Rightarrow u_i(a)\ge u_i(b) \end{aligned}$$

The utility functions now contain information about the intensity of ‘better than’, how much better an item is than another. Since we are interested in maintaining the cardinal information, rank-based methods (also known as ‘order-statistics’ methods) like the averaging Borda rank cannot be used, and the use of top-distance needs to be replaced by a direct difference between the utility and the maximum utility \(T'_i(x)=[\max _{y\in S_i} u_i(y)]-u_i(x)\) in the definition of parity. Giving meaning to such differences generally implies that only linear transformations of the form \(u'(x):=\alpha u(x)+\beta\) for \(\alpha >0\) would be admissible as transformations of u(x) that preserve the same information about parity, i.e., the underlying scale should be taken to be an interval scale.Footnote 37 Furthermore, utility functions need to be normalized to [0, 1] by the size of the sets of comparable items within each aspects. Apart from that, no changes are necessary.

The mixed case, on the other hand, poses many conceptual problems. Even when cardinal and ordinal utilities are normalized to the same interval, say the unit interval [0, 1], in a way that properly takes into account the sizes of the underlying sets of comparable items, it seems far-stretched to presume that we could simply compare them at a level by assigning a weight to each of them and using weighted sum aggregation. Or, at least there should be some substantial philosophical argument for applying this mode of aggregation in ‘better than’ comparisons. The problem is that the weights are defined for overall utilities, but it may also be the case that a is much better than b in an aspect i in a cardinal, not just in a rank-based sense of ‘much better’, whereas b is (ordinally) better than a in another aspect j. The Borda rank does take into account the position of an item, but there is no reason to believe that an ordinal rank could be compared to a cardinal utility in a meaningful way, if the difference between ordinal scales and interval scales is taken seriously.

In some cases the lexicographic structure of the evaluation might alleviate this problem if it turns out that the lexicographically most preferred aspects are all homogeneously ordinal or cardinal. However, no fully justified solution is available for the general case, other than stipulating that the proper comparison can somehow be put into the overall utility weights.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rast, E.H. The Multidimensional Structure of ‘better than’. Axiomathes 32, 291–319 (2022).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: