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1. Our discussion of cardinalizing utilities is quite similar toResnik’s (1987: 88–91).

2. The further assumptions would need to relate particular options toparticular privileged levels of utility; for instance, one would needto argue that a rational agent’s preference ordering shouldincorporate, say, a privileged zero-utility option, in which caseratios of utility distances from this option would be meaningful.

3. See Heap, et al. (1992: ch. 3), for an accessible discussion ofempirical results concerning the Allais Paradox, as well as someempirical models that have been suggested in response to theparadox.

4. Savage himself used the termconsequences rather thanoutcomes.

5. This problem of nonsensical acts is considered by some to be the mainproblem with Savage’s result (see, e.g., Joyce 1999: 108). Weraise this issue again below when discussing weaknesses ofSavage’s theory.

6. To keep things simple, we will assume that the set \(\bO\) is finite,but Savage proved a similar result for an infinite \(\bO\).

7. When the set of outcomes is infinite, the agent needs to satisfy anadditional axiom. But to keep things simple, we will assume finitelymany outcomes.

8. As some have pointed out (e.g., Zynda 2000, and Meacham &Weisberg 2011), the derived probability function over states and theutility function over ultimate outcomes need not represent theagent’s actual beliefs and desires. Perhaps the agent haspreferences over acts satisfying Savage’s axioms that are quitedisconnected from her actual beliefs and desires. Moreover,Savage’s theorem does not rule out that an agent withpreferences satisfying his axioms can be represented using analternative functional form that better matches her decision makingpsychology. These are alternative readings of the theorem. Nonethelessinference to the best explanation would seem to point in favour ofregarding the expected utility representation as uncovering the beliefand desire determinants of the agent’s preferences.

9. A result similar to Savage’s can however be obtained withoutthe Rectangular Field Assumption, in particular by adding some extrastructure to the set of prospects. See, for instance, Bradley (2007).Moreover, Gaifman and Liu (2018) have recently shown that one can getessentially Savage’s result by assuming only two distinctconstant acts.

10. Levi (1991), for instance, criticises Jeffrey’s theory forinvolving probabilities over acts, which he claims cannot be sensiblyinterpreted. Joyce (2002) and Rabinowicz (2002) however argue thatsuch probabilities are both meaningful and often necessary fordeliberation.

11. Bolker’s proof is rather complicated, and there does not seemto exist any student-friendly introduction to it.

12. This actually follows from the fact that the preferences do notdetermine a single probability function, given that the domain of theprobability function is infinite (this latter point follows from theatomlessness of \(\Omega\)). For if \(P\) and \(Q\) are probabilityfunctions on an infinite domain that agree on the ordering of allpropositions in their domain, then \(P\) and \(Q\) are the sameprobability function (Villegas 1964).