Intransitivity and vague preferences

Journal of Ethics

Volumn 11, Issue 4, 2007, Pages 377-403

  Aldred, Jonathan ^a  

^a EMMANUEL COLLEGE   (United Kingdom)

Author keywords

Preference; Rational choice theory; Rationality; Transitivity; Vagueness

Indexed keywords

EID: 34548583756     PISSN: 13824554     EISSN: 15728609     Source Type: Journal    
DOI: 10.1007/s10892-005-7977-9     Document Type: Article

References (85)

1. "I believe transitivity and reflexiveness are requirements of logic, because any comparative relation... must conform to them," J. Broome, "Deontology and Economics," Economics and Philosophy 8 (1992), p. 274.
2. See also D. Davidson, Essays on Actions and Events (Oxford: Clarendon, 1980), p. 237;
3. G. Tullock, "The Irrationality of Intransitivity," Oxford Economic Papers 16 (1964), pp. 401-406.
4. For critiques, see P. Anand, "The Philosophy of Intransitive Preference," Economic Journal 103 (1993), pp. 337-346;
5. S. Gendin, "Why Preference is Not Transitive," The Philosophical Quarterly 46 (1996), pp. 482-488;
6. R. Hughes, "Rationality and Intransitive Preferences," Analysis 40 (1980), pp. 132-143;
7. M. Philips, "Must Rational Preferences be Transitive?" The Philosophical Quarterly 39 (1989), pp. 477-483.
8. M. Bar-Hillel and A. Margalit, "How Vicious are Cycles of Intransitive Choice?" Theory and Decision 24 (1988), pp. 119-145, offer a lengthy review of many multidimensional examples of intransitive preference.
9. G. H. von Wright, The Logic of Preference (Edinburgh: Edinburgh University Press, 1963).
10. J. Broome, "Can a Humean Be Moderate?" in R. Frey and C. Morris (eds.), Value, Welfare and Morality (Cambridge: Cambridge University Press, 1993), p. 53;
11. S. Hurley, Natural Reasons (Oxford: Oxford University Press, 1989);
12. and A. Sen, "Maximization and the Act of Choice," Econometrica 65 (1997), pp. 745-779,
13. have all discussed individuation in detail, while P. Anand, Foundations of Rational Choice Under Risk (Oxford: Clarendon, 1993), pp. 103-106, has argued that any set of intransitive preferences can be given a transitive re-description, and vice-versa.
14. I thank an anonymous referee for pressing this point. See S. Rachels, "A Theory of Beneficence," unpublished undergraduate thesis, University of Oxford, 1993;
15. S. Rachels, "Counterexamples to the Transitivity of Better Than," Australasian Journal of Philosophy 76 (1998), pp. 71-83;
16. L. Temkin, "A Continuum Argument for Intransitivity," Philosophy and Public Affairs 25 (1996), pp. 175-210.
17. An interesting discussion may be found in P. Mongin, "Does Optimization Imply Rationality?" Synthese 124 (2000), pp. 73-111. Mongin is very sympathetic to the possibility of rational intransitive preferences, but implies that all existing examples of intransitivity rely heavily on their multidimensionality. Notably he assumes the coherence of the uni/multidimensional distinction without defining or defending it.
18. The following all provide examples which make such an assumption: W. Armstrong, "Uncertainty and the Utility Function," Economic Journal 58 (1948), pp. 1-10;
19. P. Fishburn, "Intransitive Indifference with Unequal Indifference Intervals," Journal of Mathematical Psychology 7 (1970), pp. 144-149;
20. J. Glover, "It Makes No Difference Whether or Not I Do It," Proceedings of the Aristotelian Society Supplement 49 (1975), pp. 171-190;
21. D. Parfit, Reasons and Persons (Oxford: Clarendon, 1984);
22. W. Quinn, "The Puzzle of the Self-Torturer," Philosophical Studies 59 (1990), pp. 79-90.
23. D. Regan, "Perceiving Imperceptible Harms," in M. Almeida (ed.), Imperceptible Harms and Benefits (Dordrecht: Kluwer Academic Publishers, 2000), pp. 4973.
24. "Bounded rationality" here follows H. Simon's usage to describe a decisionmaker whose reasoning and calculating capacities are both limited and prone to error. See H. Simon, Models of Bounded Rationality (Cambridge: The MIT Press, 1983).
25. Or some variation on this formulation. For example, the agent may be indifferent between successive developments of the magnitude specified in the DC example, unless a threshold level of development has been crossed, after which she ceases to be indifferent. Or the threshold may be defined as a proportion rather than a level, etc. These possibilities are ignored for the sake of simplicity in what follows; they either make no difference to the argument in this paragraph, or strengthen it.
26. A related point is developed in more detail in Regan, "Perceiving Imperceptible Harms," pp. 59-60, in connection with Quinn's self-torturer example.
27. Temkin, "A Continuum. Argument for Intransitivity," p. 197.
28. Temkin, "A Continuum Argument for Intransitivity," p. 200.
29. Temkin, "A Continuum Argument for Intransitivity," pp. 197-201.
30. C. Wright, "Language-Mastery and the Sorites Paradox," in R. Keefe and P. Smith (eds.), Vagueness: A Reader (Cambridge: The MIT press, 1996), p. 156.
31. Temkin, "A. Continuum. Argument for Intransitivity," p. 199.
32. Temkin, "A. Continuum. Argument for Intransitivity," p. 200.
33. Wright, "Language-Mastery and the Sorties Paradox," p. 159.
34. Wright, "Language-Mastery and the Sorties Paradox," p. 160.
35. It might be objected that "spoilt" could have a meaning involving, say, expert assessment of the viability of the land as a particular species habitat. But this sense of "spoilt" is not the one at work in the DC example.
36. Wright, "Language-Mastery and the Sorties Paradox," p. 161.
37. Wright, "Language-Mastery and the Sorties Paradox," pp. 169-170.
38. Regan, "Perceiving Imperceptible Harms," pp. 67-68.
39. Book-length treatments of vagueness include L. Burns, Vagueness: An Investigation Into Natural Languages and the Sorites Paradox (Dordrecht: Kluwer Academic Publishers, 1991);
40. T. Williamson, Vagueness (London: Routledge, 1994);
41. and R. Keefe and P. Smith (eds.), Vagueness: A Reader (Cambridge: The MIT Press, 1996).
42. For a thorough defence, see Williamson, Vagueness; he would omit the word 'definitely" in the last sentence.
43. If the interpretation is that rational preferences - what the rational agent ought to prefer - exist, but are unknowable to the agent or anyone else, then Completeness is not threatened. However, unknowable preferences appear to be incomplete preferences according to the originators of the axiom: J. von Neumann, and O. Morgenstern, Theory of Games and Economic Behavior (Princeton: Princeton University Press, 1944), p. 19,
44. describe incompleteness as "where the individual is neither able to state which of two alternatives he prefers nor that they are equally desirable;" L. Savage, The Foundations of Statistics (New York: Dover Press, 1954), p. 21, refers to "introspective sensations of indecision or vacillation, which we may be reluctant to identify with indifference."
45. Consider the instance where α= 100 so (¬S1) is "if a 100 grained collection is a heap, then a 99 grained collection is not a heap." Consider the sharpening of heap which treats all "99 or more" grained collections as heaps and all smaller collections as non-heaps. Then "an (x-1) grained collection is not a heap" will be false for α = 100. On. the proposed sharpening, the consequent of ¬S1) is false even though the antecedent is true. Thus(¬S1) is false. By the law of excluded middlfi,¬(¬S1)⇒(S1). On the proposed sharpening, (S1) is true. Since (S1) can be false if and only if false with respect to all sharpenings, it is not false for α= 100. Similar reasoning demonstrates this result for other instances.
46. Consider the instance where α = 100 and the sharpening of heap which treats all "100 or more" grained collections as heaps and all smaller collections as non-heaps. Then "an (α-1) grained collection is a heap" will not be true for a = 100. On the proposed sharpening, the consequent is false even though the antecedent is true. Thus (S1) is false.
47. The non-standard behaviour of universal and existential quantifiers in supervaluationism is a persistent source of confusion. On the supervaluational account, a universally quantified proposition can be false, even though no instance is false; an existentially quantified proposition can be true, even though, no instance is true. Thus although supervaluationism recognises no sharp boundary, it holds that (E2) is true, since for every possible sharpening, we can pick some t to make (E2) true. However if we pick t first, so to speak - pick a particular instance of (E2) - it will not be true on every possible sharpening, so not true. See M. Dummett, "Wang's Paradox," Synthese 30 (1975), pp. 301- 324.
48. D. Edgington, "Vagueness by Degrees," in R. Keefe and P. Smith (eds.), Vagueness: A Reader (Cambridge: The MIT Press, 1996), pp. 294-316, argues this point persuasively using a preference-based example.
49. F. Ramsey, "Truth and Probability," in R. B. Braithwaite (ed.), The Foundations of Mathematics and Other Logical Essays (London: Kegan Paul, 1931), pp. 156198.
50. Williamson, Vagueness, p. 124.
51. Edgington, "Vagueness by Degrees;" K. Machina, "Truth, Belief and Vagueness," Journal of Philosophical Logic 5 (1976), pp. 47-78.
52. It might be objected that normative theories of rational choice often talk of "irrationalities" by agents, such, as imperfect calculating capacity, even though it is readily conceded that no human will be perfect in this respect. Thus the ignorance springing from epistemic vagueness is not improperly described as irrational. Reply: (i) While a "transitive imperfect calculator" agent is typically held to be "boundedly rational," I know of no standard theory which grants an intransitive perfect calculator the same status. Rather, the latter type is labelled irrational - the status of intransitivity is not the same as that of imperfect calculation, (ii) On the epistemic theory, universal ignorance is not an incidental empirical observation, but essential to the account. Vagueness could not be understood without human failure, while there are agents with perfect calculating powers in some situations (I have in mind the rare people who can perform complex integrations in their head).
53. Quinn, "The Puzzle of the Self-Torturer," Temkin, "A Continuum Argument for Intransitivity."
54. M. Almeida, (ed.) Imperceptible Harms and Benefits (Dordrecht: Kluwer Academic Publishers, 2000).
55. J. Broome, Ethics Out of Economics (Cambridge: Cambridge University Press, 1999), p. 9.
56. Parfit, Reasons and Persons;
57. Quinn, "Self Torturer;" G. Schumm, "Transitivity, Preference and Indifference," Philosophical Studies 52 (1987), pp. 435-437.
58. This view of decision theorists is explicitly stated in both. Mongin, "Does Optimization Imply Rationality?" p. 80,
59. and Anand, Foundations of Rational Choice Under Risk, p. 26.
60. R. Luce, and H. Raiffa, Games and Decisions (New York: Wiley, 1957)
61. and P. Fishburn, Utility Theory for Decision-Making (New York: Wiley, 1970) are two classics in decision theory which endorse intransitive indifference, although Fishburn at least went on to accept intransitive strict preference too.
62. Anand, Foundations of Rational Choice Under Risk, p. 27.
63. Schumm, "Transitivity, Preference, and Indifference," p. 435.
64. This presentation of the Schumm model is due to W. Rabinowicz, "Money Pump with Foresight," in M. Almeida (ed.), Imperceptible Harms and Benefits (Dordrecht: Kluwer Academic publishers, 2000), pp. 123-154.
65. Schumm, "Transitivity, Preference, and Indifference," p. 436.
66. See Mongin, "Optimization," p. 80; Rabinowicz, "Money Pump with Foresight," p. 128.
67. See the next sub-section for a statement of the argument.
68. Schumm, "Transitivity, Preference, and Indifference," p. 437.
69. The argument was initiated by D. Davidson, J. McKinsey, and P. Suppes, "Outlines of a Formal Theory of Value, I," Philosophy of Science 22 (1955), pp. 140160; they developed a suggestion by Frank Ramsey.
70. Put differently, standard money pumps require the assumption that the agent's preference between any two successive stages in the cycle is strong enough not to be reversed by her payment to move along a stage. The analogous assumption here is that the agent's preference between end states of the cycle is strong enough not to be reversed by the sum of small payments she has received along the way. There may be a problem with these assumptions (or relatedly, with the assumption that each such payment is arbitrarily small), but this does not disturb the analogy between Dodgy Developers' activities and standard money pumps S. Hansson, "Money Pumps, Self-Torturers and the Demons of Real Life," Australasian Journal of Philosophy 71 (1993), pp. 476-485, has shown formally that the only difference between money pumps based on indifference cycles and standard cyclic preference pumps is that the former make an additional mild (and uncontroversial) monotonicity assumption.
71. Rabinowicz, "Money Pump," p. 124.
72. F. Schick, "Dutch Bookies and Money Pumps," The Journal of Philosophy 83 (1986), p. 118.
73. Rachels, "Counterexamples to the Transitivity of Better Than," p. 80.
74. There is a large literature on the money pump but two important recent contributions are made by R. Cubitt, and R. Sugden, "On Money Pumps," Games and Economic Behavior 37 (2001), pp. 121-160, and Rabinowicz, "Money Pump." After a very formal, analysis, Cubitt and Sugden conclude that the money pump objection to intransitivity does not stand. Rabinowicz's position is more complicated, but his ultimate conclusion also rejects the money pump argument.
75. See also J. Aldred, "The Money Pump Revisited," Risk, Decision and Policy 8 (2003), pp. 59-76.
76. Quinn, "The Puzzle of the Self-Torturer," Rachels, "Counterexamples," Temkin, "A Continuum Argument."
77. Rachels, "A. Set of Solutions to Parfit's Problems," Nous 35 (2001), pp. 214238.
78. Anand, Foundations of Rational Choice Under Risk, p. 56.
79. Anand's ideas are anticipated by T. Schwartz, "Rationality and the Myth of the Maximum," Noûs 6 (1972), pp. 97-117.
80. It is true that a well-known principle, variously known as the Chemoff or contraction condition or Property a, allows us to deduce binary preferences from ternary ones. However, the contraction condition is closely linked to transitivity (formulations of both, axioms vary, but in most versions contraction implies transitivity), so the appeal probably begs the question. Moreover, A, Sen,"Is the Idea of Purely Internal Consistency of Choice Bizarre?" in J. Altham and R. Harrison (eds.), World, Mind and Ethics (Cambridge: Cambridge University Press 1.995), has developed a number of examples and arguments suggesting the contraction condition is implausible.
81. Rachels, "Counterexamples to the Transitivity of Better Than," p. 83.
82. J. Griffin, Well-Being (Oxford: Clarendon, 1986), p. 96.
83. Rabinowicz, "Money Pump," pp. 147-150;
84. Parfit, Reasons and Persons, p. 496499;
85. J. Sobel, "Cyclical Preferences and World Bayesianism," Philosophy of Science 64 (1997), pp. 42-73.