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Show Chapter IV. The Utility-Maximizing Model of Rationality: Formal Interpretations 154 and I. Because, unlike the indifference relation, the equivalence relation does not assign value to its terms (not even equal value), it cannot show this. Then the consequent of (A2), here rewritten as (A2'c) [(t F) ≡ (t G)] . [(~t I) ≡ (~t H)] follows as a statement about the independence of the connective relations among F, G, H, and I from the content of the ethically neutral toss-up belief on the truth of which I am willing to bet either way for the same stakes. But it does not show the independence of the equality of the value intervals between F and G and between H and I from that belief. The equality of these intervals depend upon my indifference between (iii") and (iv"). Because indifference is an intensional relation that ranges over the objects of value within its scope, it is not permissible to substitute or logically manipulate the terms of this relation with impunity as Ramsey does and can do with impunity the terms of the equivalence relations set out in (A2). Within the scope of an intensional operator, probabilistically identical toss-up beliefs are not automatically intersubstitutable. For example, I may strictly prefer peaches to pears given an expected 50% chance that a random coin-toss will come up heads. But I may be indifferent between them given an expected 50% chance of rain, because my apprehension about the weather ruins my appetite. So it remains moot whether the value intervals that separate F, G, H and I remain equal relative to some toss-up belief different than s. This raises the question of what it is that axioms (A3) and (A4) stipulate the transitive consistency of. Without an unproblematic derivation of the equality of value intervals FG, GH, and HI that accommodates the intensionality of these values and the relations among them, we lack explicit guidelines for understanding how value options can be equivalent (A3) and how the intervals among them therefore can be equal (A4). We have just seen that an unexplicated transition from indifference-talk to equivalence-talk does not suffice. Then the ordinal ranking of F, G, H and I yields no guidelines for ascertaining what it would mean to speak of the transitivity of "equivalent" value options (A3), nor what it would mean to speak of the transitivity of "equal" value intervals (A4). It appears that the familiar, extensional interpretation of the transitivity relation is the only one available (I examine the Jeffrey-Bolker solution to this problem in Volume II, Chapter III.6.2). To see why this does not suffice, try replacing the equivalence relation in (A3) with the indifference relation abandoned in the move from (A1) to (A2). Using "≈" to mean "is indifferent to," (A3) becomes (A3') If A ≈ B and B ≈ C, then A ≈ C. © Adrian Piper Research Archive Foundation Berlin |
