Philosophy & Public Affairs 31.3 (2003) 272-279
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Defending Transitivity against Zeno's Paradox
Ken Binmore and Alex Voorhoeve
Recently, Larry Temkin and Stuart Rachels have offered a new argument against the transitivity of the relationship 'all things considered better than'. 1 This argument, in its various guises, invokes our intuitions about our preferences over different bundles of pleasurable or painful experiences with different durations, which, it is argued, will typically be intransitive. This article defends against this argument the orthodox view that the relation 'all things considered better than' should be regarded as transitive by showing that Temkin and Rachels are mistaken in supposing that a preference relation satisfying their assumptions must be intransitive. It makes clear where the argument goes wrong by showing that it is a version of Zeno's paradox of Achilles and the tortoise.
Their argument centers around two kinds of counterexamples to the transitivity of 'all things considered better than', one involving painful and the other involving pleasurable experiences of differing duration. Since Temkin and Rachels both offer an example of painful experiences, we begin by discussing this case in Section I. Section II explains why the argument fails. Section III explains why the case of pleasure is of the [End Page 272] same form, and can be solved in the same way. Section IV contains some general remarks on counterexamples to the transitivity of 'all things considered better than'.
I. The Argument
Temkin and Rachels's pain counterexample rests on the following three claims, which they hold to be true: 2
Claim 1: For any unpleasant or 'negative' experience, no matter what the intensity and duration of that experience, it would be better to have that experience than one that was only a little less intense but that lasted much longer. 3
Claim 2: There is a finely distinguishable range of unpleasant or 'negative' experiences ranging in intensity from mild discomfort to extreme agony.
Claim 3: No matter how long it must be endured, mild discomfort is preferable to extreme agony for a significant amount of time.
Temkin and Rachels believe that these claims taken together contradict the transitivity of 'all things considered better than'. They invite us to imagine first a lengthy life that contains an amount of excruciating torture of significant but relatively short duration. Let us call this combination of torture and the time it must be endured T0. It follows from the first claim that one would prefer a life containing T0 to an otherwise identical life that contains T1, where T1 represents slightly less intense torture for a much longer period of time. Claim 2 then allows us to apply the first claim repeatedly with a slightly lower intensity of torture and a longer time period to generate a chain of preferences, which Temkin and Rachels argue runs from T0 (excruciating torture for a short period of time) at one end to TMILD (a mild pain for a very long time) at the other. Temkin envisages the mild pain as a hangnail, Rachels as a slight headache. Each member of this chain is preferred to its successor. By transitivity, T0 is preferred to TMILD. But by claim 3, TMILD is preferred to T0, so that claims 1, 2, and 3 generate an intransitive preference. [End Page 273]
Temkin and Rachels deduce far-reaching conclusions from their purported demonstration that it may be reasonable for people to have intransitive preferences. Rachels advocates abandoning what he calls "maximizing theories" for evaluating the goodness of states of affairs in favor of his own "quasi-maximizing theory" that embraces intransitivity. 4 Temkin quotes Derek Parfit to the effect that the argument leads to a "general skepticism about practical reasoning," and concludes that "whatever we say, in the end, about my arguments, they may require us to seriously rethink our understanding of the good, moral ideals and the nature of practical reasoning." 5 We believe no such drastic measures are required, since, as we show in the next section, claims 1...