If MONO-MON8 hold of <W, N , P, >>, then there exists at least one countably additive probability function Bel on P and a real-valued function Des* on the atomic elements w o fP , whose associated conditional expected utility Des on P is such that for all P, Q, CP U - p ) G P,
(i) Des(P) =
Yw
Bel(w\P).Des*(w)(ii) Des{P U ~P) = 0
(iii) Des(P) > Des(Q) iff P > Q
Furthermore, the pair <Bel, Des> is unique up to a fractional linear transformation
Note that the representation involves two utility functions: Des* is defined only on the atomic elements o f P , while Des is defined for all P G P and characterised in terms o f B el and Des*.
The uniqueness properties o f this representational system are quite different than those we find in other CEU theorems. Neither Bel, Des*, nor Des are unique; in stead, the pair <Bel, Des> is unique up to a fractional linear transformation. Des is normalised such that Des(R U ~P) = 0 and Des(R) = 1, for some proposition R
satisfying MON1. Let in f be the greatest lower bound o f the values assigned by
Des, and let sup designate the least upper bound. Finally, let X be a parameter falling between -M inf and -Msup. Then the fractional linear transformation <Bel\, Desy>
o f <Bel, Des> corresponding to X is given by:
BelfP ) = Bel{P).{ 1 + XDes(P))
D esfP) = Des(P).((\ + X) / (1 + XDes(P))
Interestingly, fractional linear transformations o f a <Bel, Des> pair can alter not only the absolute values that B el assigns to propositions, but also their relative val-
ues; i.e., different possible representations o f exactly the same system o f prefer ences will sometimes disagree regarding which o f two propositions has should be assigned a higher credence. More generally, an agent’s preferences on this kind of monoset framework do not typically determine a unique relative credence ordering > b on T . Jeffrey suggested that it would be possible to pin down a unique Hel if > b were treated as a primitive relation on par with >, with its own set o f conditions (e.g., in his 1974, 1983). Joyce (1999, 138ff) proved that this is possible.
6.2.2 Critical discussion
Jeffrey’s monoset theorem is one o f the best-known amongst philosophers. Amongst other disciplines, however, the monoset framework is often considered problematic. As Fishbum puts it,
Although well known in certain philosophical circles, Jeffrey’s work is infrequently cited, and by implication not widely known, in other disciplines that share the legacy of preference and decision theory ... A casual search of works on the foundations of decision and relational measurement in the fields of psychology, economics, statis tics and management science indicates that if Jeffrey’s work is mentioned at all, it is likely to be in reference to The Logic o f Decision, and then only to note that it pro poses a theory of decision that differs from traditional paradigms. (1994, 136)
There are, consequently, very few representation theorems based on an ontologi- cally similar framework. Two recent exceptions to this trend can be found in (Bradley 1998, 2007) and (Ahn 2008), and as noted above, Armendt’s (1986) the orem is ontologically similar to Jeffrey’s system in that it takes preferences to be defined on a set o f propositions and the possible roulette lotteries that may be formed thereupon. To keep the discussion brief, I will focus my criticisms on The orem 6.3— the main points to be discussed apply equally to the other theorems just mentioned.
As far as characterisational representationism is concerned, Theorem 6.3 seems to take us several steps in the right direction. In particular, it neither appeals to act- functions nor lottery-functions— two bugbears which we have seen create problems
for the multiset theorems considered so far. Furthermore, the domain o f its prefer ence relation is not limited to some obscure class of entities (such as infinite con junctions o f counterfactuals or lotteries upon lotteries), but instead seems capable
o f encapsulating everything towards which we could have mentalistic preferences. O f course, a mentalistic construal o f > means that Theorem 6.3 fails to satisfy the naturalistic desideratum (5), but we have seen that the standard strategies for trying to formulate a theorem around the behavioural notion o f preference lead to far worse concerns for characterisational representationism. Finally, Theorem 6.3’s 'Bel and Ves are defined on precisely the same domain, a feature not shared by any o f the multiset theorems we have considered so far (desideratum (2a)).
Theorem 6.3 does, however, have some limitations; these I will note below, though first I want to briefly discuss one characteristic o f Theorem 6.3 that I don’t take to be especially problematic— in particular, the theorem’s relatively weak uniqueness conditions. These are often cited as a cause for concern, as though char acterisational representationism must be based upon a theorem which comes with (at least) the Standard Uniqueness Condition. But it’s difficult to see why this should be so.
There are at least two (not mutually exclusive) strategies by which a proponent o f characterisational representationism might attempt to deal with Theorem 6.3’s weak uniqueness results. First, one can appeal to information which goes beyond agents’ (actual or counterfactual) preferences. This further information can be used to narrow down the range o f potential interpretations whenever a representation theorem does come with strong uniqueness conditions. For example, if the theo rem ’s “Bel function is non-unique, a principle like Charity might be used to constrain the set o f available Bel representations down to uniqueness (§4.2). Second, where a theorem supplies us with a restricted set o f possible Bel and Ves representations, we might take the entire set as a model o f the agent’s credences and utilities. After all, Theorem 6.3 does carry the implication that there is a unique set o f <Bel, Ves> pairs (each related to the others by a fractional linear transformation) such each such pair jointly T-represents > on T . Perhaps, then, that unique set— the ‘represen tor’— might be used to jointly represent the agent’s credences and utilities: roughly, whatever is true o f every Bel in the set is true o f the agent’s credence state (and likewise for their utilities). So, for instance, if every Bel in the representor always
assigns a higher value to Pthan to Q, then the agent’s credence in Pis higher than her credence in Q. Something close to this suggestion was briefly discussed in
§5.2.4, and the idea was raised by Jeffrey in his (1983).
Neither o f these two strategies comes without cost, o f course. If the former is adopted, then the characterisation o f credences and utilities mustappeal to infor mation that goes beyond the agent’s preferences; this may be considered too much for some die-hard advocates o f a very strict form o f preference functionalism. On the other hand, if the latter strategy is adopted, then the very intuitive picture o f an agent as an expected utility maximiser must be sacrificed for a rather more complex model involving the interaction total credence and utility states modelled by sets of Pel and Ves functions. Nevertheless, neither o f these costs seems like a strong enough reason to reject the possibility o f basing characterisational representation- ism on something like Theorem 6.3.
If there is a serious problem with Theorem 6.3, it relates to whether its Pel and Ves functions (or sets thereof) can serve as accurate models o f an ordinary agent’s credences and utilities (desideratum (2)). For one thing, the Pel associated with
Theorem 6.3 is always a probability function, which puts limits on the kinds of credence states that it can represent— though some o f the issues here depend on whether the set
W
is taken to be a set o f possible worlds. If it is, then Pelis limited to the representation o f probabilistically coherent agents— and ipso facto incapable o f representing the average person. The same is true o f any representor set con structed solely from probability functions: for instance, every probability function T r built on a space o f possible worlds will assign 0 to impossible propositions, 1 to necessary propositions, and satisfies the property that if P \- Q,then Tr(P) < Pr(Q).It may, however, be possible to avoid this implication by letting
W
be composed o f both possible and impossible worlds (Nolan 1997), although taking this route may lead to other concerns (see, e.g., Bjerring 2013). Another problem, however— and one that an appeal to impossible worlds w on’t help with— is that if Pelis to be a probability function, then its domainT
must be closed under (at least finite) dis junctions, yet it may be too much to ask o f ordinary agents that they have credencestowards everydisjunction P v Q which can be formed from the propositions Pand Q towards which they do have credences (desideratum (2c)). Worse still, in Jef frey’s system,
P - N
is required to be a bottomless algebra, so Pel and Ves mustbe defined on a collection o f ever-increasingly more specific propositions— propo sitions which quickly become far too specific for any ordinary agent to contem plate.84 And finally, Ves(P) always equals the Be/-weighted average utility of the various different ways that P might come true. It is implausible that ordinary agents’ utilities are so consistently rational in this way.
I do not consider these problems to be especially devastating, at least if the task is to develop a version o f characterisational representationism aimed solely at ide ally rational agents. However, I think we can do better— in Chapter 8 , 1 will develop a theorem which is ontologically very similar to Theorem 6.3, but with much less restricted Pel and Des functions and a more plausible representation overall. Before getting to that, though, we need to conclude our review o f the representation theo rems currently on offer for characterisational representationism with the very first such theorem to have been developed: Frank Ramsey’s.
84 Domotor (1978) proves a theorem similar to Bolker’s for the case where T is finite. He relies, however, on a particularly strong condition that he calls projectivity, and his uniqueness condition is weaker than Theorem 6.3’s.