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¹⁷ If (but only if) the set of outcomes is sufficiently rich, one can instead take the i-cardinal structure to be given by an ordering of pairs (‘the difference between x ₁ and x ₂ is at least as great as the difference between x ₃ and x ₄’), and still end up with numerical scales that are unique up to positive affine transformation; see the discussion of ‘intrapersonal difference comparability’ in Bossert, W. and Weymark, J. A., ‘Utility in Social Choice’, Handbook of Utility Theory, vol. 2: Extensions, ed. Barbera, S., Hammond, P. and Seidl, C. (Heidelberg, 1977), pp. 1099–1177 Google Scholar, at 1127–8, and references therein.

¹⁸ We also need to impose requirements of mutual consistency between the i-ordinal and i-cardinal (resp., i-cardinal and co-cardinal) structures for a given quantity. Consistency between i-ordinal and i-cardinal structure: if a ≃ _i b, c ≃ _i d, e ≃ _i f, g ≃ _i h ∈ X and $C_{i}(a,c,e,g)=r\in \mathbb{R}$ , then C_i (b, d, f, h) = r. Consistency between i-cardinal and cocardinal structure: if C_i (a, b, c, d) = C_j (e, f, g, h), and if in addition (a, b; i) ~ (e, f; j), then (c, d; i) ~ (g, h; j).

¹⁹ But not inevitable: cf. the ‘extended preferences’ approach to grounding interpersonal comparisons, discussed in e.g. Harsanyi, Rational Behavior, secs. 4.2–4.4; J. Broome, ‘Extended Preferences’, Preferences, ed. Fehiga and Wessels, pp. 271–87; Adler, M., Well-being and Fair Distribution: Beyond Cost–Benefit Analysis (Oxford: 2012), ch. 3Google Scholar; H. Greaves and H. Lederman, ‘Extended Preferences and Interpersonal Comparisons of Well-being’ (forthcoming in Philosophy and Phenomenological Research, 2016).

²⁰ For further discussion of weighted utilitarianism and interpersonal comparisons of utility in the context of Harsanyi’s theorems, see, e.g. Mongin and d’Aspremont, ‘Utility Theory and Ethics’, sec. 5.2. For Harsanyi-style theorems that aim to establish unweighted utilitarianism via the imposition of an additional axiom of ‘anonymity’, see Mongin and d’Aspremont, ‘Utility Theory and Ethics’, Proposition 5.3; d'Aspremont, C. and Mongin, P., ‘A Welfarist Version of Harsanyi's Aggregation Theorem’, Justice, Political Liberalism, and Utilitarianism (Cambridge, 2008), pp. 184–97CrossRefGoogle Scholar), Theorem 7.2.

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³² The difficulty of the question has often been noted in the literature on prioritarianism: see e.g. Broome, J., Weighing Goods (Oxford, 1991)Google Scholar; Parfit, D., ‘Another Defence of the Priority View’, Utilitas 24 (2012), pp. 399–440 CrossRefGoogle Scholar; Greaves, H., ‘Antiprioritarianism’, Utilitas 27 (2015), pp. 1–42 CrossRefGoogle Scholar.

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³⁴ For discussion of various results connecting separability conditions to additive representations and of the range of possible applications of those results, see e.g. Blackorby, C., Primont, D. and Russell, R. R., ‘Separability: A Survey’, Handbook of Utility Theory: vol. 1: Principles (Heidelberg, 1998), p. 49 Google Scholar, sec. 5; D. von Winterfeldt, W. Edwards, et al. (1986). Decision Analysis and Behavioral Research (Cambridge), pp. 331–4; Broome, Weighing Goods.

³⁵ For overviews of attempts to articulate a causal principle as whole or part of metasemantic theory, see K. Neander, ‘Teleological Theories of Mental Content’, The Stanford Encyclopaedia of Philosophy (Spring 2012 edition), <http://plato.stanford.edu/archives/spr2012/entries/content-teleological/>; Rupert, R. D., ‘Causal Theories of Mental Content’, Philosophy Compass 3.2 (2008), pp. 353–80CrossRefGoogle Scholar. On the ‘charity plus eligibility’ programme, see especially Lewis, D., ‘New Work for a Theory of Universals’, Australasian Journal of Philosophy 61.4 (1983), pp. 343–77CrossRefGoogle Scholar; Lewis, D., ‘Putnam's Paradox’, Australasian Journal of Philosophy 62.3 (1984), pp. 221–3CrossRefGoogle Scholar.

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⁴⁰ As noted already by W. Vickrey, ‘Measuring Marginal Utility by Reactions to Risk’, Econometrica (1945), pp. 319–33.

⁴¹ Mongin, ‘Impartiality, Utilitarian Ethics, and Collective Bayesianism’.

⁴² M. Fleurbaey and P. Mongin, ‘The Utilitarian Relevance of the Aggregation Theorem’ (n.d., unpublished manuscript).

⁴³ For valuable discussions, I am grateful to Ted Sider, Robbie Williams, and participants in the 2014 Conference on Rational Choice and Philosophy at Vanderbilt University, especially Christian List and John Weymark. Thanks also to an anonymous referee for extremely helpful comments and suggestions.

⁴⁴ Harsanyi’s own presentations of (especially) the Impartial Observer result are rather informal. The formulation outlined here is close to that provided by Weymark, ‘A Reconsideration of the Harsanyi–Sen Debate on Utilitarianism’.

⁴⁵ I.e. the probability that π assigns to extended alternative (A, i) is given by the product π _X (A) · π _I (i).