Millian Superiorities

Utilitas

Volumn 17, Issue 2, 2005, Pages 127-146

  Arrhenius, Gustaf ^a,b   Rabinowicz, Wlodek ^a,b  

^a STOCKHOLM UNIVERSITY   (Sweden)

^b LUND UNIVERSITY   (Sweden)

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EID: 85008535023     PISSN: 09538208     EISSN: 17416183     Source Type: Journal    
DOI: 10.1017/S0953820805001494     Document Type: Article

References (25)

1. Object types should be distinguished from object kinds. Two wholes composed of objects of the same type can be said to be of the same kind. Thus, a whole composed of m e objects is of the same kind as a whole composed of n e objects, for any m, n, and e.
2. Note that, according to this definition, e may be weakly superior to e′ even when e′ is better than e. If this sounds unnatural, one might always add to the definition an additional requirement that e must be better than e′ in order to be weakly superior to it. However, for simplicity's sake, we prefer to work with a more austere definition of weak superiority.
3. Mill's own label for this kind of superiority is 'superiority in quality'. According to his preference test of superiority, what decides the superiority issue among pleasures is that persons who are 'competently acquainted' with both kinds of pleasures would prefer a single pleasure of one kind to 'any quantity of the other pleasure which their nature is capable of' (J. S. Mill, Utilitarianism, in On Liberty and Other Essays, ed. J. Gray (Oxford, 1991 [1866]), p. 138).
4. In J. Griffin, Well-Being: Its Meaning, Measurement, and Moral Importance (Oxford, 1986), p. 85, James Griffin seems to have something like weak superiority in mind when he considers a form of 'Discontinuity' in which 'enough of A outranks any amount of B'. On the other hand, superiority is similar to what Griffin (p. 83) calls 'Trumping', which 'takes the form: any amount of A, nomatter how small, is more valuable than any amount of B, no matter how large'. For a different interpretation of Griffin's distinction, however, see appendix 3.
5. We are indebted to Roger Crisp for pressing this point.
6. ∗. Therefore, it follows from the Archimedean axiom that for some k, k+1 e′-objects are better than a whole composed of e together with k e′-objects. Consequently, if adding e′ to any whole never worsens the value of that whole, we reach the desired conclusion that, for some m, (m=k+1), m e′ objects are better than a single e.
7. W. Rabinowicz, 'Ryberg's Doubts about Higher and Lower Pleasures-Put to Rest?', Ethical Theory and Moral Practice 6 (2003).
8. J. Ryberg, 'Higher and Lower Pleasures-Doubts on Justification', Ethical Theory and Moral Practice 5 (2002).
9. Cf. R. Crisp, 'Utilitarianism and the Life of Virtue', The Philosophical Quarterly 42 (1992)
10. J. Riley, 'On Quantities and Qualities of Pleasure', Utilitas 5 (1993)
11. J. Riley, 'Is Qualitative Hedonism Incoherent?', Utilitas 11 (1999)
12. D. Parfit, 'Overpopulation and the Quality of Life', Applied Ethics, ed. P. Singer (Oxford, 1986). It is arguable that infinite or lexical superiority also figures in the section on 'Discontinuity' in Griffin's Well-Being. There, he suggests that when 'enough of A outranks any amount of B' (as in the cases of weak superiority), 'we have positive value [B] that, no matter how often a certain amount [of that value] is added to itself, cannot become greater than another positive value [enough of A], and cannot, not because with piling up we get diminishing value or even disvalue (though there are such cases), but because they are the sort of value that, even remaining constant, cannot add up to some other value' (Griffin, Well-Being, p. 85, our italics). This can be read as a suggestion that, in those cases, one positive value is infinitely or lexically greater than another. However, as is clear from this quote, Griffin also recognizes cases in which the reason for 'Discontinuity' does not have anything to do with infinite (or lexical) value differences. As for Mill's own views on the matter, we leave that issue to the experts.
13. It does not really matter in the present context whether one interprets the superiority in question as infinite or as lexical. On the lexical view, a superior object carries a value that may be finite but is of a higher order than the value of an inferior object. These two constructions-the infinitistic and the lexical one-are closely related interpretations of the same idea. In particular, both imply that if the first element of the object sequence is superior to the last one, in the requisite sense, then at some point in a decreasing object sequence, a corresponding superiority relation must set in between the adjacent elements. (Parenthetically, however, one should add that the infinitistic and the lexical interpretations of superiority are not equivalent. In fact, the former lends itself to some counter-intuitive implications that the latter avoids. To see this, consider lotteries. On the standard view, the value of a lottery that gives a chance p to a prize e equals p times the value of e. In other words, the value of a lottery is its expected value. Consequently, lotteries that give different (positive) chances to an infinitely valuable prize have all the same value, independently of chance differences: they are all infinitely valuable. No such counter-intuitive conclusion is forthcoming on the lexical view. On that view, a lottery that assigns a positive chance to a higher-order prize may be lexically better than any lottery with a prize of a lower order, but at the same time still worse than lotteries that assign to the higher-order prize a higher chance.)
14. Ryberg, 'Higher', p. 418.
15. Cf. G. E. Moore, Principia Ethica (Cambridge, 1993 [1903]), sects. 18-21, et passim. To be more precise, Moore does assume a form of additivity when he suggests that 'the value on the whole' is the sum of (i) the values of the parts plus (ii) 'the value of the whole, as a whole'. (The latter may be either positive or negative.) But it can be argued that this form of additivity is a purely arithmetical construct. His 'value of the whole, as a whole' could simply be interpreted as the arithmetical difference between 'the value on the whole' and the sum of the values of the parts, independently considered. Moore himself points out that the value of a whole, as a whole, may be 'expressed' as such a difference, but he seems to ascribe to it some independent significance.
16. Moore was not the first philosopher to make this point. Another standard reference is F. Brentano, The Origin of our Knowledge of Right and Wrong (London and Henley, 1969 [1989]).
17. Qualification: If e′ is a whole in which e′ is replaced by e, the restriction on the replacement is that e and e′-minus-e′ are disjoint: No part of the former is a part of the latter.
18. Cf. D. Parfit, Reasons and Persons (Oxford, 1984), sect. 137. His application of this idea to the value of populations comes from T. M. Hurka, 'Value and Population Size', Ethics 93 (1983). For a discussion, see G. Arrhenius, Future Generations: A Challenge for Moral Theory (Uppsala 2000), ch. 4.
19. The proof above assumes that the relation 'at least as good as' is complete, which is a rather exacting requirement. What if completeness is not assumed? Well, even in the absence of completeness, we can prove a variant of Observation 3. Definition 3: An object e is minimally superior to an object e′ iff for some number m, no whole composed of e′-objects, however large, is better than m e-objects. Observation 4: Suppose that at-least-as-good-as is a transitive relation. If the first element in a finite sequence of objects isminimally superior to the last element, then there must exist some element in that sequence that is minimally superior to its immediate successor. For the proof, see appendix 2. For a similar result in the context of population axiology, see Arrhenius Future Generations, secs. 3.2 and 10.3-4. Note that weak superiority entails minimal superiority, but not vice versa. Still, just as it is the case with these stronger relations, minimal superiority is incompatible with the Archimedean axiom.
20. Cf. the Quantity Condition discussed in Arrhenius, Future Generations, pp. 54-6, which captures this intuition in a population context. It is shown that this condition, together with some other weak conditions, implies the Repugnant Conclusion.
21. On this view, then, being weakly superior to an object is not sufficient for beingmuch better than the object in question. In fact, it is not necessary for being much better either. It is easy to construct a case in which, in a descending sequence e, e′, e′, (i) the value difference between the first element and the second one is larger than that between the second element and the third, (ii) the second element is weakly superior to the third, but (iii) the first element is not weakly superior to any of the elements that follow. While a sufficient number of e′-objects may be better than any number of e′-objects and e may be much better than e′, there still may be no such number of e-objects that cannot be outweighed by a sufficiently large number of e′-objects.
22. This article is a significantly revised and expanded version of G. Arrhenius and W. Rabinowicz, 'On Millian Discontinuities', in Patterns of Value, vol. 1, ed. Wlodek Rabinowicz and Toni Rønnow-Rasmussen (Lund, 2003). We are indebted to Roger Crisp for a helpful comment that led us to Observation 1 and to John Broome for useful ideas as to how to extend superiority relationships between objects to analogous relationships between attributes (cf. appendix 3, co-authored by Broome and Rabinowicz). Rabinowicz's work on this article was supported by a research grant from The Bank of Sweden's Tercentenary Foundation and was completed during his stay at the Swedish Collegium for Advanced Study in the Social Sciences in Uppsala.
23. It is arguable that James Griffin had something like these relations between attributes in mind when he distinguished between 'Trumping' and 'Discontinuity'. In the case of Trumping, any positive amount of one attribute, A, no matter how small, is preferable to any amount of the other attribute, B. In the case of Discontinuity, on the other hand, a sufficient amount of A is preferable to any amount of B. (Cf. Griffin Well-Being, pp. 83 and 85.) However, Griffin's definitions are framed in terms of amounts in which an attribute could be present rather than in terms of possible improvements with respect to a given attribute, as we have done. The latter approach is preferable if one wants to allow for the possibility of attributes that are negatively rather than positively value-relevant. For such attributes it is better if they are present in smaller rather than larger amounts. Furthermore, avoiding reference to amounts is advisable if one wants to leave room for attributes that cannot be measured on a ratio scale.
24. Consider the following quote from Griffin: 'The mistake here seems to be to think that certain values-liberty, for instance-as types outrank other values-prosperity, for instance-as types. Since values, as types, can vary greatly in weight from token to token, it would be surprising to find this kind of discontinuity at the type- or at least fairly abstract type-level. It is much more promising to look at less abstract types, or at tokens'. It appears that what Griffin might have had in mind in this passage is that it is more realistic to expect that superiority relations between relatively abstract attributes obtain relative to particular reference-points (i.e. on the level of 'tokens') rather than globally.
25. However, the qualification comes at a cost. If an attribute A is such that no object in the domain admits of A-improvements (which will be the case if changes in A always require simultaneous changes in some other attribute), then, given the qualification, it follows that A is (vacuously) globally weakly superior to every other attribute. If no object admits of B-improvements either, then A is (vacuously) globally weakly superior to B, and vice versa. Thus, with the qualification, weak superiority stops being an asymmetric relation. Still, these problems do not arise in domains in which for every attribute there exist some objects that could be improved with respect to that attribute. Furthermore, the same problems exist for global superiority. If no objects can be A-improved or B-improved, because changes in each of these attributes always are correlated with changes in other attributes, then, vacuously, A is globally superior to B and B is globally superior to A.