Parity, interval value, and choice

Ethics

Volumn 115, Issue 2, 2005, Pages 331-350

  Chang, Ruth ^a  

^a NONE

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EID: 14544277558     PISSN: 00141704     EISSN: None     Source Type: Journal    
DOI: 10.1086/426307     Document Type: Article

References (22)

1. Ruth Chang, "The Possibility of Parity," Ethics 112 (2002): 659-88.
2. Joshua Gert, "Value and Parity," Ethics 114 (2004): 492-510. All subsequent page references in parentheses are to this article.
3. Another recent attempt to do away with parity is mooted by Nien-he Hsieh, "Equality, dumpiness, and Incomparability," Utilitas (in press), who argues that a "clumpy" understanding of value allows us to treat cases of parity as cases of equality. But Hsieh's understanding of 'being equally good' as "belong[ing] to the same clump of value" (p. 12) is saddled with a dilemma. According to Hsieh, A and B can belong to the same clump of value even though something better than A, A+, also belongs to the same clump of value. This means that even though A+ is better than A, which is equally good as B, A+ is not better than B but is equally good as B. This is not equality as we know it. To avoid this result, Hsieh seems to suggest that we must relativize evaluative comparisons to a degree of precision. Thus, A and B are equally good to degree of precision pi, A+ is better than A to degree of precision p2, and A+ and B are equally good to degree of precision pi (or perhaps p3). The trouble with this suggestion is that it commits us to denying the inferential links between evaluative comparisons that proceed with respect to the same covering consideration. If Mozart is better than Salieri with respect to creative genius, and Salieri is better than Talentlessi, a rotten sculptor, with respect to creative genius, it seems to follow that Mozart is better than Talentlessi with respect to creative genius. This is so even though the degree of precision according to which a comparison of the creative genius of musicians proceeds is different from the degree of precision according to which a comparison of a musician and a sculptor proceeds.
4. I say that these "appear" to be definitions of parity because although Gert says that he will offer definitions of parity that explain it away, he never explicitly says what these definitions are.
5. Moreover, Gert's first formulation of parity arbitrarily emphasizes one alternative over the other when in fact there is symmetry: not just one, but either of two items on a par might be improved without thereby becoming better than the other.
6. As Gert writes, "This paper will defend the trichotomy thesis, at least in one important sense: it will hold that any other positive value relations that we might wish to make use of can be defined in terms of the three traditional relations" (p. 493).
7. Sometimes I get the sense that what Gert really objects to is a reification of parity as something out there in the world. But that is no part of the claim that there is parity; there "is" parity in just the same sense in which there is betterness, worseness, and equality in value. What is at issue is parity's status vis-à-vis the usual three relations, not its ontological status.
8. Gert gives a choice-theoretic gloss of these intervals as representing the rationally permissible strengths of preference one might have for an item, but this is to import Humean assumptions about value that Gert recognizes we should ignore if we are to give his model its widest possible scope. My discussion of interval representation is neutral between different ways in which we might understand in what the value of an item consists.
9. It is important to emphasize that when Gert suggests that there are more relations beyond the usual three plus parity, he is not claiming that there is any relation with logical properties different from what I call parity; his claim, rather, is that what I call parity can be "divided up" into many different relations. Whether he is right about this waits on an account of how value relations are properly individuated, In any case, Gert's only defense of the claim depends on the details of his model, and, as we will see, his model is problematic.
10. If a configuration represents the first interval as better than the second, then its "reverse" will swap the intervals from left to right and represent the first interval as worse than the second. The reverse of betterness is worseness and vice versa. Since equality and parity are both symmetric, their reverses will represent equality and parity, respectively.
11. Gert goes on to say that a case like case 6 in which the intervals, though the same, have a very narrow range is a case of "rough equality" (pp. 506-7). But it is clear that which relation holds between two items should not turn on the size of the intervals representing them if all relational aspects of the intervals are held constant. Gert is misled by the fact that intervals with small ranges approach a degenerate interval, i.e., a point (with items represented by the same point being equally good). But approaching this degenerate interval and being identical to it are quite different matters.
12. In e-mail correspondence, Gert suggested that the proof can be resisted on the grounds that an item is not as good as itself. Without going into the plausibility of rejecting the reflexivity of equality, we can alter the proof to be one about an item and its twin: take any interval range and let the values of an item and its twin be given by that range. The item and its twin are equally good. Again, the Range Rule must be rejected.
13. It will not help to suggest, as Gert did in e-mail correspondence, that there is some way of making the improved item better than its unimproved ancestor beyond that sanctioned by the Range Rule since the Range Rule is supposed to provide a model of all relations of comparability.
14. If, e.g., in fig. 1, we flip configuration 3 upside down, we get the reverse of configuration 4. Duality implies that if the first interval of configuration 3 is better than the second, then the first interval of the reverse of configuration 4 must be worse than the second interval in that configuration. That is, if configuration 3 represents betteress, the reverse of configuration 4 must represent worseness. (And if the reverse of configuration 4 represents worseness, then configuration 4 must represent betterness.)
15. In his "Modelling Parity," draft manuscript, Wlodek Rabinowicz cites a theorem employed by Peter Fishburn (and brought to his attention by Erik Carlson) that shows that interval modeling has insufficient "dimensionality" to cover a configuration that we should expect sometimes to hold when there is parity. Dimensionality is a technical concept having to do with the least number of linear orderings whose intersection is the partial ordering. Using the same configuration that Fishburn relies on to reject interval modeling on the basis of dimensionality, I argue that interval modeling should be rejected on the basis of the somewhat more intuitive condition of pair dictatorship. I am grateful to Rabinowicz for bringing Fishburn's result to my attention. See Rabinowicz, "Modelling Parity"; Erik Carlson, "Incompatibility and the Measurement of Value" (draft manuscript, Department of Philosophy, Uppsala University); and Peter C. Fishbum, Interval Orders and Interval Graphs - a Study of Partially Ordered Sets (New York: Wiley, 1985), pp. 78 ff., as cited in Rabinowicz. Rabinowicz goes on to offer his own interesting suggestion as to how parity should be modeled. Rabinowicz's model, however, does not undermine the thought that parity is a genuinely distinct fourth value relation.
16. In his "Modelling Parity," draft manuscript, Wlodek Rabinowicz cites a theorem employed by Peter Fishburn (and brought to his attention by Erik Carlson) that shows that interval modeling has insufficient "dimensionality" to cover a configuration that we should expect sometimes to hold when there is parity. Dimensionality is a technical concept having to do with the least number of linear orderings whose intersection is the partial ordering. Using the same configuration that Fishburn relies on to reject interval modeling on the basis of dimensionality, I argue that interval modeling should be rejected on the basis of the somewhat more intuitive condition of pair dictatorship. I am grateful to Rabinowicz for bringing Fishburn's result to my attention. See Rabinowicz, "Modelling Parity"; Erik Carlson, "Incompatibility and the Measurement of Value" (draft manuscript, Department of Philosophy, Uppsala University); and Peter C. Fishbum, Interval Orders and Interval Graphs - a Study of Partially Ordered Sets (New York: Wiley, 1985), pp. 78 ff., as cited in Rabinowicz. Rabinowicz goes on to offer his own interesting suggestion as to how parity should be modeled. Rabinowicz's model, however, does not undermine the thought that parity is a genuinely distinct fourth value relation.
17. In his "Modelling Parity," draft manuscript, Wlodek Rabinowicz cites a theorem employed by Peter Fishburn (and brought to his attention by Erik Carlson) that shows that interval modeling has insufficient "dimensionality" to cover a configuration that we should expect sometimes to hold when there is parity. Dimensionality is a technical concept having to do with the least number of linear orderings whose intersection is the partial ordering. Using the same configuration that Fishburn relies on to reject interval modeling on the basis of dimensionality, I argue that interval modeling should be rejected on the basis of the somewhat more intuitive condition of pair dictatorship. I am grateful to Rabinowicz for bringing Fishburn's result to my attention. See Rabinowicz, "Modelling Parity"; Erik Carlson, "Incompatibility and the Measurement of Value" (draft manuscript, Department of Philosophy, Uppsala University); and Peter C. Fishbum, Interval Orders and Interval Graphs - a Study of Partially Ordered Sets (New York: Wiley, 1985), pp. 78 ff., as cited in Rabinowicz. Rabinowicz goes on to offer his own interesting suggestion as to how parity should be modeled. Rabinowicz's model, however, does not undermine the thought that parity is a genuinely distinct fourth value relation.
18. See Derek Parfit, Reasons and Persons (Oxford: Oxford University Press, 1984), p. 431, and unpublished work; and James Griffin, Well-Being (Oxford: Clarendon, 1986), pp. 80-81, 96-98.
19. See Derek Parfit, Reasons and Persons (Oxford: Oxford University Press, 1984), p. 431, and unpublished work; and James Griffin, Well-Being (Oxford: Clarendon, 1986), pp. 80-81, 96-98.
20. Compare Gert's claim that the rationality of choice is not simply a function of an agent's valuation of the alternatives. According to Gert, a rational agent might recognize that objects take a certain range of value-points, but to preserve "consistency," she must choose in a way that does not involve her taking those objects to have particular value-points that could lead to the charge of "inconsistency" in choice (p. 504). In short, to avoid the value pump problem, Gert adds a free-floating constraint that an agent not allow herself to be a value pump, while proponents of parity build such a constraint into the proper valuation of items as on a par.
21. Of course, as I have argued elsewhere, the value pump problem arises also when there is incomparability. But we can distinguish two sorts of value pump puzzles: those in which the value pump is created by choices delivered by a choice function, and those in which it is created by the failure of a choice function to deliver a correct choice. My focus here is on the puzzle arising when the choice function is not silent. See my "Introduction," in Incommensurability, Incomparability, and Practical Reason, ed. Ruth Chang (Cambridge, Mass.: Harvard University Press, 1997).
22. The assumption that configuration 6 is the only case of equality can be replaced by the weaker assumptions of the transitivity and substitutability of equality. The assumption that configuration 0 is a case of betterness can be dropped, in which case we should also allow the converse of the Pareto Interval Rule to hold. The duality assumption can be weakened so as not to involve the implication listed as fact 1. If, however, duality is dropped altogether, we no longer have interval modeling in the sense that both Gert and I have in mind, i.e., as modeling by a relaxation of the reals where direction has significance.