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Show Rationality and the Structure of the Self, Volume I: The Humean Conception 147 A sorities paradox can be generated for any judgment involving a quantitative concept, and a fortiori for any assignment of weights and probabilities to preference alternatives, whether they fall within the normal or the infinitesimal range. Therefore any Allais assignment must fall within the series of premises [(i), (ii), (iii), … (iv)-(v)] in (a), or [(i'), (ii'), (iii'), …(iv')-(v')] in (b). So it is hard to imagine on what special grounds, or according to what special criterion, the extreme assignments on which the Allais Paradox relies can be excluded from an agent's preference ranking. On the contrary: other things equal, certain preference alternatives (excluding death and taxes) may enter the outermost edges of an agent's field of choice with probability assignments that decrease in magnitude with their temporal distance from the agent, and, so long as their weights do not decrease to zero, increase in aggregate value as the agent moves forward in time to meet them. My preferences for having an apple for breakfast and residing in Berlin exactly twenty years from today would be of this kind. Alternatives that enter the field at the most temporally remote outer edges may well bear, at that point, the infinitesimal probability assignments on which Allais-style paradoxes thrive. They may remain preference alternatives nevertheless. 1.6.2. Sorites, Cyclicity and the vN-M Axioms I have just argued that Allais assignments must be admissable in an agent's preference ranking - thereby empirically disconfirming the independence axiom via the Allais Paradox. But the same Pro-G or Contra-G arguments that generate sorities paradoxes also generate cyclical rankings that structurally violate all of the von Neumann-Morgenstern axioms, and indeed several others. We have seen above that, by contrast with the vagueness built into the quantitative gradience of weight and probability assignments, whether or not Percy's ordinal ranking of the options available to him is consistent or not is (at least on the face of it) not a matter of degree but rather of classical logic. If G is among Percy's live options at a particular moment, then at that moment it either is or is not the case that Percy prefers H to F, F to G, and so H to G. However, that a sorites paradox can be generated in either case results from the fact that G's negligible aggregate value may figure in an argument on either side of this disjunction. G's aggregate value in case 1.6.1.(2) favors the Pro-G argument to the same extent that it favors the Contra-G argument. G's aggregate value would need to increase considerably in degree in order to shift the balance decisively in favor of the former over the latter, or else decrease to 0 in order to shift it in the opposite direction. In the absence of any such incremental increase or decrease, it both is and is not the case that Percy prefers H to F, F to G and so H to G. From these unresolvable ruminations on case 1.6.1.(2) we can now easily generate formal violations of the other condition of the vN-M weak ordering © Adrian Piper Research Archive Foundation Berlin |
