A continuum argument for intransitivity

Philosophy and Public Affairs

Volumn 25, Issue 3, 1996, Pages 175-210

  Temkin, Larry S ^a  

^a NONE

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EID: 1042287762     PISSN: 00483915     EISSN: 10884963     Source Type: Journal    
DOI: 10.1111/j.1088-4963.1996.tb00039.x     Document Type: Article

References (40)

1. More generally, any relation R will be transitive, if and only if, for any a, b, and c, if aRb and bRc, then aRc. "Taller than" is a standard example of a transitive relation, since if Andrea is taller than Becky, and Becky is taller than Claire, then Andrea is taller than Claire. By contrast, "being the mother of" is not a transitive relation, since Andrea's being the mother of Becky, and Becky's being the mother of Claire, does not entail - and in fact is incompatible with - Andrea's being the mother of Claire.
2. Philosophy & Public Affairs 16, no. 2 (Spring 1987): 138-87
3. A Theory of Justice (Cambridge, Mass.: Harvard University Press, 1971), p. 30.
4. A classic source for these and other fascinating psychological results is Judgments under Uncertainty: Heuristics and Biases, ed. Daniel Kahneman, Paul Slovic, and Amos Tversky (Cambridge: Cambridge University Press, 1982).
5. A pioneering article on this topic was Amos Tversky's "Intransitivity of Preferences," Psychological Review 76, (1969): 31-48.
6. This view was voiced to me on different occasions by Thomas Nagel, Derek Parfit, and Thomas Scanlon. I don't know whether Nagel still holds this view; Parfit does, but Scanlon does not.
7. Weighing Goods (Oxford: Basil Blackwell, 1991), p. 11.
8. Rachels's original example is contained in his unpublished Philosophy, Politics and Economics thesis "A Theory of Beneficence" (Oxford University, 1993). Rachels was a student of Parfit, and in many respects his example is similar to examples Parfit and I have developed and discussed over the years. But Rachels's example is a clear improvement on our examples. It is cleaner and more compelling. Rachels has modified his original example, as well as offered other arguments for intransitivity, in two important unpublished papers, "Reconceiving Better Than" and "Counterexamples to the Transitivity of Better Than."
9. I have presented these claims to many audiences. Although, together, they yield a counterintuitive conclusion, each, individually, has been overwhelmingly accepted.
10. Fred Feldman has sent me a long list of such explanations, many of which are quite ingenious and sophisticated. Although I don't find these explanations convincing, they are worthy of careful consideration, and their appeal is certainly understandable.
11. See Chapter Three of Derek Parfit's Reasons and Persons (Oxford: Oxford University Press, 1984);
12. also the striking results in Kahneman, Slovic, and Tversky's Judgments under Uncertainty: Heuristics and Biases.
13. I am grateful to Peter Unger for this example.
14. Note, I do not believe that any amount of torture, no matter how short, must be worse than any amount of a hangnail, no matter how long. Nor does my argument commit me to this. It might well be that two seconds worth of torture would be better than many years of a hangnail, and it is almost certainly true that two nanoseconds of torture would be better than many years of a hangnail. Correspondingly, there may be perfectly transitive orderings from outcomes involving very short amounts of torture to one's involving very long amounts of a hangnail. I don't deny this, nor do I need to. All I need to establish is one case of intransitivity, and for this it is enough if torture that lasts long enough, would be worse than a hangnail of vastly longer (and even perhaps any) duration. My model for this view is noted later in the text, and note 18.
15. Readers who still think it will take too long to get from extreme torture to a hangnail may want to consider variations of my example. Here is one some find convincing. Imagine a scale of "uncomfortable" experiences, ranging from a level of 1, a hangnail, to a level of 100, extreme torture. Suppose a moderately uncomfortable limp is an 11 - significantly worse than a mere hangnail, but not nearly as bad as extreme torture. Start with a choice between A, level 100 discomfort for 2 days, or B, level 80 discomfort for 4 days. B's discomfort is 20 percent less than A's but lasts twice as long. Many think A better than B. Next compare B to C, where C stands to B as B stands to A. C is 20 percent less intense, level 64, but lasts twice as long, 8 days. Again, many think B better than C. The tenth choice would be between J, discomfort of level 13.4 for 1024 days, and K, discomfort of level 11 for 2048 days. Again, many would think J better than K. Given these rankings, transitivity entails that A is better than K. Is it? Given our assumptions, A involves extreme torture for 2 days, K involves a moderately uncomfortable limp for 2048 days, or 5.6 years. Which would you choose for your child - that for the next 5.6 years they suffer an uncomfortable limp, or for 2 days they suffer extreme torture? (Here, as always, we assume there are no relevant side-effects. Other children do not mercilessly ridicule your child as a "gimp"; a pill, or hypnosis, removes all memories of the torture; etc.) I would choose the limp over the torture for someone I loved. And I would do so because I think it the better alternative. But I would also choose A over B, B over C, . . . , and J over K. This example involves an imaginable range of pains, and finite graspable times. Here, as before, I reject transitivity.
16. See page 184.
17. Reasons and Persons, p. 388.
18. Parfit's arguments concerning the Repugnant Conclusion are fascinating and, together with related considerations he presents - especially the Mere Addition Paradox - first sparked my interest in intransitivity. See Part Four of Reasons and Persons, and my "Intransitivity and the Mere Addition Paradox."
19. Parfit's seminal publication on this topic was "Future Generations: Further Problems," Philosophy & Public Affairs 11, no. 2 (Spring 1982): 113-72.
20. See pp. 85-89 of Well-Being (Oxford: Oxford University Press, 1986). In these pages, Griffin offers a striking argument that is in some respects similar to mine. However Griffins argument is offered in favor of "discontinuity," and I suspect Griffin would resist the radical conclusion I am suggesting regarding its implications for intransitivity. In an important, but ultimately unsatisfying note, Griffin traces his example to Parfit's Repugnant Conclusion, and suggests how he would try to avoid it (note 27, pp. 338-40).
21. There are many disanalogies between pennies and grams, on the one hand and pleasures and pains, on the other. Unfortunately, it isn't easy to pin down the precise disanalogies that matter here. Factors that seem relevant include the storability and transferability of pennies and grams, which allow for the possibility of pennies and grams to be literally brought together and compounded across people and time in a way that pleasures and pains are not. In thinking about these issues, I find it useful to consider examples like the following. Suppose I was deeply committed to leaving $500,000 to a future descendent. Suppose I were willing to make a great personal sacrifice to accomplish this goal. Regarding my goal, I should clearly be indifferent between making the great sacrifice in exchange for: one person putting a lump sum of $500,000 into an interest-free account today; one person putting $15 per day every day for the next 90 years into an interest-free account; or each ot 33,333 people putting $15 into an interest-free account today. Now suppose I was equally concerned to relieve anyone from suffering intense pain for several years, and was willing to make a great personal sacrifice to achieve this end. Is it clear that I should be equally willing to make the same great sacrifice in order to achieve the seemingly different goal of relieving one person from suffering a much milder pain for a significantly longer time or in order to prevent many others from experiencing a minor pain for a short time? I think not. To repeat: pleasures and pains are not transferable and compoundable across people and time the way pennies are, and the way it seems they would need to be to reject claim 3.
22. It is because torture of a second or less might have a disvalue close to 0, and many years of a hangnail might have a disvalue close to 1, that I reject the view that any amount of torture must be worse than any amount of a hangnail. But on my view, once torture lasts long enough it will have a disvalue greater than 1, at which point, on this model, it will be worse than any duration of a hangnail.
23. This section draws on material contained in "Intransitivity and the Mere Addition Paradox," as well as in "Weighing Goods: Some Questions and Comments," Philosophy & Public Affairs 23, no. 4 (Winter 1994): 350-80.
24. Unfortunately, doing so adequately lies well beyond this article's scope. I began the task in my "Intransitivity and the Mere Addition Paradox," in particular in my discussion of Intrinsic Aspect and Essentially Comparative views of moral ideals, and have briefly revisited the issue in several articles since. I hope, one day, to make further progress in a much larger project tentatively titled "Rethinking the Good, Moral Ideals, and the Nature of Practical Reasoning."
25. I shall spare the reader further details of this example. I trust it is clear how the example should be filled out given the relevant assumptions.
26. It may occur to some readers of this article that my schematic explanation of intransitivity is questionable because it appears to work too well. If my explanation is correct, such readers may argue, then it should be easy to generate many instances of uncontroversial intransitivities that fit the pattern I have suggested, yet I don't supply such examples. Also, a reader may further argue that if one accepts this explanation, then intransitivity might no longer seem so troubling in the way that it now does. Let me briefly respond. First, from the fact that there may be some cases of intransitivity which fit the pattern, it does not follow that there must be many such cases or that they should be readily apparent. The circumstances that render different factors relevant, or of varying significance, for comparing different alternatives may be rare, subtle, or obscure. Having said that, however, I think one can generate a fair number of intransitivities that fit my pattern. I did not do so in this article, mainly because I thought it would be best carefully to present and examine perhaps the single strongest example; partly, because doing so adequately would have significantly lengthened an already long article; and also, because I present a number of other examples elsewhere. Some of these are already in print (see my "Intransitivity" and my "Weighing Goods"), others will appear shortly (see my contribution to Well-Being and Morality: Essays in Honour of James Griffin, ed. Roger Crisp and Brad Hooker [Oxford University Press, forthcoming]). Also, I agree that once my explanation is accepted, the intransitivity of "all things considered better than" should no longer seem so puzzling. And I am pleased that indeed many readers of my work on intransitivity now find the idea plausible. Still, my explanation ultimately relies on a conception of moral ideals (an essentially comparative view) that is opposed by an alternative conception (an intrinsic aspect view) that is natural, plausible, and extremely powerful. Since the alternative conception is widely held, and entails that "all things considered better than" must be transitive, this helps to explain why most people at first find my claims about intransitivity perplexing, and why it may take a while to convince them and others. (See "Intransitivity.") Finally, even if one accepts my account of intransitivity - so that the notion is no longer difficult to understand - there remain reasons of the sort noted in Sections II and VII to find the issue deeply troubling. This may help account for the negative reaction most have to the claim that "all things considered better than" may be intransitive.
27. I am grateful to the editors of Philosophy & Public Affairs for suggesting this as a possible response, since I suspect some version of it is likely to occur to many readers.
28. For a related, rather surprising, discussion applying a continuum argument for intransitivity to the abortion debate see my contribution to the volume honoring James Griffin referred to in note 22.
29. This is the basis of the classic dictum de gustibus non est disputandum ("in matters of taste there can be no dispute").
30. Note, I am not suggesting, as some may think, that if one could demonstrably prove which flavor of ice cream were best, then the inconsistency of people's preferences would be unproblematic. First, if there were a fact of the matter about which flavors were better than others, that fact might well involve a transitive ordering, in which case someone's intransitive preferences would fail to accurately reflect reality. Second, even if, somehow, there were a fact of the matter about how flavors compared that was accurately reflected by intransitive preferences, there would still be serious problems, as noted in Sections II and VII, resulting from those intransitivities. It is not particularly troubling to insist that someone "get their preferences in order" - say, by preferring chocolate to vanilla rather than vice versa - if there is no fact of the matter about which is better. However, it would be problematic to insist, say, that one prefer chocolate to vanilla if, in fact, vanilla was better than chocolate.
31. What is normatively and theoretically disturbing is the notion that there could be a fact of the matter about how alternatives compare that was accurately reflected by intransitive preferences. This is why we needn't be too worried about intransitive preferences in cases where there is no fact of the matter about how alternatives compare, or by the mere existence of intransitive preferences in cases where there is such a fact. If we reject the view that collective preferences determine which outcomes are better, then the fact that collective preferences are often intransitive, for understandable reasons having nothing to do with the goodness of the outcomes themselves, need not itself especially worry us (theoretically, as opposed to practically). Absent considerations of the sort this article presents - supporting the possibility that intransitive preferences might, in fact, accurately reflect reality - intransitive collective preferences would merely serve to remind us how easily collective preferences can go astray in ranking alternatives.
32. There are as many betterness relations as there are objects to be compared in terms of different criteria. Cucumbers can be better and worse; so can cars, pens, tomatoes, and paintings. It would be interesting if the "better cucumber than," or "better pen than," or "better painting than" relations were intransitive. But ultimately, questions about such betterness relations do not matter as much as questions about the moral betterness relation. The moral betterness relation concerns the fundamental normative question of what ought to be the case. This, in turn, impacts the fundamental normative questions of what ought I to do, and what kind of person ought I do be. Thus, if the moral betterness relation is intransitive, this can deeply affect our understanding of the normative universe, including the nature and extent of our obligations. Section VII addresses this issue further; but much more work needs to be done on this crucial topic.
33. Obviously, I part company with those who think morality is a human invention and "merely" a matter of social convention. On such a view, it is always an option to change our morality - by changing our agreements or conventions - whenever doing so best suits our purposes. See, for example, John Mackie, Ethics: Inventing Right and Wrong (Harmondsworth: Penguin, 1977);
34. and Gilbert Harman, The Nature of Morality (Oxford: Oxford University Press, 1977).
35. In "Supererogation and Obligation," Journal of Philosophy 82 (1985): 118-38.
36. Technically, an ordering obtains for any set that is ranked according to a transitive relation, R; that is, for all members of the set, x, y, and z, if xRy, and yRz, then xRz. R is a ranking relation for the set if and only if R is transitive. So, technically, any relation that is intransitive is not a ranking relation, and does not yield an ordering of the set to which it applies. Thus, transitivity is a property of orderings or rankings, and intransitivity is not. (I mention these facts, because although the following discussion does not require a technical interpretation of the notion of ranking, someone unaware of them may find the following needlessly confusing.) Incidentally, a ranking relation may be complete - if for all members of the set either xRy, or yRx, or both; or incomplete - if some members of the set are not rankable vis-à-vis each other; that is, if for some members of the set, x and y, neither xRy nor yRx. If R is complete, it yields a complete ordering of the set; if R is incomplete, it yields a partial or quasi-ordering.
37. This lesson of my arguments against transitivity was first suggested to me - though not necessarily endorsed - by my colleague Baruch Brody. Many others have since echoed this suggestion.
38. I raise a similar worry about John Broome's defense of the sure-thing principle in "Weighing Goods: Some Questions and Comments."
39. In discussion. I have worries about this expression, but they need not concern us here.
40. I am reminded here, and sympathetic to, Thomas Nagel's methodological remark in the preface to Mortal Questions, "Given a knockdown argument for an intuitively unacceptable conclusion, one should assume there is probably something wrong with the argument that one cannot detect." But I take some comfort in his immediately following qualifying remark, "though it is also possible that the source of the intuition has been misidentified," as well as in his preceding comment, "I believe one should trust problems over solutions, intuition over arguments, and pluralistic discord over systematic harmony. Simplicity and elegance are never reasons to think that a philosophical theory is true: on the contrary, they are usually grounds for thinking it false" (Cambridge: Cambridge University Press, 1979), p. x.