conditional desirability

RICHARD BRADLEY

CONDITIONAL DESIRABILITY

ABSTRACT. Conditional attitudes are not the attitudes an agent is disposed toacquire in event of learning that a condition holds. Rather they are the compo-nents of agents’ current attitudes that derive from the consideration they giveto the possibility that the condition is true. Jeffrey’s decision theory can be ex-tended to include quantitative representation of the strength of these components.A conditional desirability measure for degrees of conditional desire is proposedand shown to imply that an agent’s degrees of conditional belief are conditionalprobabilities. Rational conditional preference is axiomatised and by applicationof Bolker’s representation theorem for rational preferences it is shown that condi-tional preference rankings determine the existence of probability and desirabilitymeasures that agree with them. It is then proven that every conditional desir-ability function agrees with an agent’s conditional preferences and, under cer-tain assumptions, every desirability function agreeing with an agent’s conditionalpreferences is a conditional desirability function agreeing with her unconditionalpreferences.

KEY WORDS: Conditional attitudes, Beliefs, Desires, Rationality, Conditionalprobability, Conditional desirability, Decision theory, Representation theorem,Conditional preference

INTRODUCTION

We are capable of expressing a wide variety of attitudes of a con-ditional nature. I believe, for instance, that if I stay up all night, Iwill be unable to concentrate the next morning; I want to go outtomorrow, but if it’s going to rain then I would rather to stay athome; if it’s fish for dinner tonight, I would prefer white wine tored; I prefer to be in New England if the leaves are changing colour,than in Florida if it’s very humid; and so on. Characteristically weexpress conditional attitudes by means of conditional sentences andone might hope that the right treatment of the latter would yieldan understanding of the conditional attitudes. Unfortunately, how-ever, there is too much controversy at present over the semantics ofconditionals for this approach to be relied on.

Theory and Decision47: 23–55, 1999.© 1999Kluwer Academic Publishers. Printed in the Netherlands.

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24 RICHARD BRADLEY

Nonetheless, conditional attitudes deserve study. Knowing howthey are related to one another and to unconditional beliefs anddesires would help us to anticipate and understand the behaviourof others and to refine our own. Because our decisions are generallysensitive to our beliefs, and because the latter may be uncertain orunstable, conditional attitudes often play an essential role in rationaldecision making. Suppose, for instance, that I am unsure as to whatI want to do tomorrow, but sure that if it’s sunny I will want to go tothe beach, and sure that if it is not, I will want to do something else.Then I should arrange things so that, should it be sunny, I can go tothe beach and should it not be, that I am not committed to doing so.Or if unsure as to which restaurant to go to, but having conditionalpreferences highly sensitive to their affordability, I would be advisedto determine what my current financial situation is before attemptingto make a decision.

An agent’s conditional beliefs and desires should not be identifiedwith what she is currently disposed to believe or desire in the eventthat the condition were realised. It might be that I currently desirethat if I get sick, I will continue to go to work, but also the case thatwere I to get sick, I would desire not to go to work because I wouldfeel too bad. Similarly it is possible that one both believes that ifone drinks too much one will drive badly, and be such that wereone to drink one would believe oneself able to drive well. So one’sconditional attitudes can differ from the attitudes one would take ifthe condition in question were realised.

There is a dispositional theory of conditional attitudes that ismuch closer to the truth. The view could be expressed thus: to be-liefe thatA if B is to be conditionally disposed to believeA in theevent of believingB, and to desire thatA if B is to be disposed todesireA in the event of believingB, and to desire thatA if B is to bedisposed to desireA in the event of believingB.1 But this view toorequires some qualification. Consider again the case where I desirethat if I get sick I will continue to go to work and suppose that Iwas to learn that I was sick by starting to feel bad. Then again thefeeling bad may cause me to lose my desire to work. Or a case inwhich I desire that if I win the sweepstakes we should go out andcelebrate immediately, but where it is so unlikely that I will win that,were I in fact to do so, it would be such a shock that I would lose all

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CONDITIONAL DESIRABILITY 25

desire to celebrate. These cases show that to say that someone hasa conditional belief or desire with respect to some condition is notquite the same as saying what they would believe and desire shouldthey learn that the condition was true.

What these examples trade on in essence is the possibility ofchanges in an agent’s attitudes which are caused by the realisationof the condition, but which are not rational consequences of hercurrent beliefs and desires. This suggests that a conditional beliefor desire thatX given Y is not a counterfactual attitude toX (anattitude that one would take), but an attitude that onepresentlytakesto the truth ofX on the assumption or hypothesis thatY . Or to put itdifferently, they are attitudes that derive from the current considera-tion one gives to the epistemic possibility ofY , not from the causaleffect on one’s attitudes of learning thatY . The current attitudes thatone has on the hypothesis that some condition is true explain one’sdisposition to change one’s beliefs and desires in particular ways inthe event of learning that the condition holds true. But one may haveother, non-rational dispositions to change one’s beliefs and desireswhich are not part of one’s current conditional attitudes.

The relation between the conditional attitudes and the consid-eration given to various epistemic possibilities will be made moreprecise later on, but I offer the following as a preliminary character-isation: to have aconditional beliefin X if Y is to believeX andYto the degree that one believesY . And to have aconditional desirefor X if Y is to desireX andY to the degree that one believesY . Forinstance, suppose I doubt that I will win at poker tonight because Idon’t think I will be lucky, but don’t doubt that I will win any morethan I doubt that I will be lucky and win. Then I may be said tohave a conditional belief that I will win at poker, given that I will belucky. For if I did not believe that I would win in the event of beinglucky, I should doubt that I will win to greater extent that I doubtthat I will be lucky.

Now to understand better what is meant here by degrees of beliefand desire, one must recognise that beliefs and desires have not onlya direction (their content), but also a strength or force. For instance,suppose that I desire to get some fresh air but not to get wet, so that adecision as to whether to go for a walk or not will depend on whetherit is going to rain in the next hour or not. Often I will not be sure as

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26 RICHARD BRADLEY

to which will be the case and, hence, partially disposed to go for awalk (or to make it true that I go for a walk) and partially disposed tostay indoors. Then we may say that I desire to some degree to go fora walk and to some degree to stay indoors, these degrees of desirebeing measures of the strength of my desire as directed towards andaway from taking a walk.

In this example, a belief that it is not going to rain in the nexthour would dispose me to take a walk and a belief to the contrarywould dispose me to stay indoors. So, all other factors being equal,I may be said to currently believe that it will rain to the extent thatI am actually disposed to stay indoors (to the degree that I desire todo so.) And to believe the opposite to the extent that I am disposedto go out. What I am disposed to do overall depends on the relativestrengths of my beliefs in the possibility of it raining and of it notdoing so, or on the degree to which my belief in directed towardsone possibility or another. In this sense my degrees of belief in thetwo directions are determinants of my disposition or desire to take awalk.

On the standard view of degrees of belief as probabilities, thecase in which an agent has a conditional belief inA givenB hasprecise representation as one in which the agent’s degree of beliefinAB equals her degree of belief inB. No such precision is affordedmy characterisation of conditional desire. For while one can speakof full belief or certainty, and therefore interpret a degree of beliefas a proportion of full belief, there is no corresponding state of fulldesire. Desire is always a relative matter: one desires one thing moreor less than another without any one thing counting as the fullydesired one. (To desire thatX is simply to desireX more than thestatus quo or whatever other threshold is assumed in a particularcontext.) Since there can be no unit of full desire, degrees of desirecannot be scaled like probabilities. The notion of desiringA to thedegree that one believesB to be true is correspondingly rough.

My present aim is not, however, to make the suggested notionof conditional desire more precise (given the relational nature ofdesire I doubt that this is possible), but to extend the decision theo-retic treatment of belief and desire to the corresponding conditionalstates. What decision theory gives us in regard to the former isa description (in the form of a pair of measures) of those formal

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properties of a rational agent’s degrees of belief and desire that areimplied by their role in determining a rational agent’s preferences.In similar fashion I will propose a measure for conditional desirabil-ity, explore its relation to the standard measure of conditional beliefand then prove a theorem showing that both of these measures areimplied by the conditional preferences of a rational agent.

In this paper I rely on the version of decision theory formulatedby Richard Jeffrey.2 Its distinguishing characteristic is the ‘news-value’ theory of desire: the view that one should preferZX toY justin case one prefers to get the news thatX is true than to get the newsthatY is: i.e., just in case the expected state of the world given thetruth ofX is preferred to the expected state of the world given thetruth of Y . This treatment of desirability is controversial, but I willmake no argument here in favour of it as opposed to the alternativesfound, for instance, in Savage’s work or in causal decision theory.Nor will I do much by way of comparing my treatment of condi-tional attitudes with corresponding work in these other traditions.In any case, differences are essentially inherited from the underly-ing treatment of desirability and do not derive especially from ourrespective treatments of conditionality. (Indeed, I suspect a causaldecision theorist could satisfactorily adapt my theory by systemati-cally replacing my references to desirability with reference to somecausality sensitive alternative.)

I will use capital letters to denote both propositions and propo-sitional variables, context serving to make it clear which it is.T

is the tautologous proposition andF the contradictory proposition.The following symbols will refer to the respective propositionaloperators or agent-proposition relations.∼ negation∨ disjunction∧ conjunction, which is otherwise represented by writing two propo-

sitions without any space between them= equivalence>,> preferred to, at least as preferred as≈ indifferent betweenThroughout� will be complete atomless Boolean algebra of propo-sitions from whichF has been removed.

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Appendix 1 should be consulted for statements of Bolker’s Ex-istence Theorem, Jeffrey’s Uniqueness Theorem and some conse-quences of them that will be assumed throughout. The equivalenttheorems for conditional desirabilities are to be found in Appendix2. Any longish proof of my own has been placed in Appendix 3.

DESIRABILISTIC DEPENDENCE

I argued that one’s conditional attitudes of belief and desire are notthe attitudes one would have if one came to believe the conditiontrue, but components of one’scurrentattitudes to their truth. In thisrespect conditional beliefs and desires are special limiting cases ofa general sensitivity of agents’ degrees of belief and desire to theconsideration they give to various alternative epistemic possibilities.When I claim that if it rains I will get wet, but that if it doesn’t I willstay dry, I express the fact that I currently believe that I will getwet precisely to the degree that I believe that it will rain and believethat I will stay dry to the degree that I believe that it will not. Ingeneral, however, I may only believe to some degree that I will getwet if it rains and to some degree that I will get wet if it doesn’t.And my overall degree of belief in the possibility of getting wet willbe a resultant of my respective degrees of conditional belief in thatpossibility given that it will or will not rain.

A degree of conditional belief in someX given someY is ameasure of just that component of one’s degrees of belief inX

that derives from the consideration that one gives to the epistemicpossibility thatY is true. Rational degrees of conditional belief arestandardly measured by conditional probability functions, the latterbeing defined as follows. If Prob is any probability measure, thenthe conditional probability function, Prob( |Y), is such that:

CONDITIONAL PROBABILITY. If Prob(Y ) 6= 0,

Prob(X|Y) = ProbXY/ProbY

An agent’s degrees of conditional belief are conditional probabilitieson the pain of inconsistency: if an agent’s degrees of conditionalbelief do not equal the ratios of her degrees of unconditional belief,

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a Dutch book can be made against her, in which a sequence of con-ditional bets is guaranteed to leave the bookie with a profit wheneverthe condition is realised.3

Let Des(X|Y) be read as the ‘conditional desirability ofX givenY ’ Analogously to the conditional probability, it will serve as a mea-sure of the strength of an agent’s conditional desires – the degree towhich the agent (presently) desires the truth of some propositionX

if some other propositionY is the case or on the assumption thatY istrue. By this we mean that it measures that component of her currentdegrees of unconditional desire deriving from the consideration shegives to the possible truth ofY (and not the degrees of desire theagent would have wereY true).

There are two ways in which one might expect an agent’s degreeof desire for someX to be influenced by the supposition thatYis true. The first is the effect of the truth ofY on the subjectivevalue to her of the news thatX is true. If it will be sunny tomorrow,for instance, the news that I am a scheduled for a trip to the beachwill be judged vary favourably, because beaches are pleasant placesto be when the weather is good. Not so, on the other hand, if itwill rain tomorrow. So in general one might expect the conditionaldesirability ofX givenY to depend on the desirability ofXY .

The second is the effect of the truth ofY on the degree to whichthe agent believesX. For instance, if I know that we always go to thebeach when its sunny, then the subsequent news that we will indeeddo so will be of little interest to me. The assumption that it willbe sunny, as it were, robs the beach trip of its news-value. So theconditional desirability ofX givenY depends on the probabilisticdependence ofX onY .

The limiting case of this probabilistic dependence is one in whichthe truth of a propositionY implies that of propositionX or, moregenerally, where Prob(X|Y) = 1. Here the news thatY is true car-ries the news thatX is and there is no more news inXY than inY(a situation characterised behaviourally by the agent’s indifferencebetween the truth ofXY and that ofY ). In this case, once it is giventhat Y is true, there can be no interest in learning thatX is. So ifwe make the natural move of equating all cases in which nothing islearnt, the conditional desirability measure can be normalised withrespect to the unconditional desirability measure on tautologies i.e.

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30 RICHARD BRADLEY

we may stipulate that Des(X|Y) should equal DesT wheneverXcarries no news over and above what is carried byY . We state thislatter condition more formally as:

NORMALITY CONDITION. If Prob(X|Y) = 1,

then Des(X|Y) = DesT .

The normality condition is not in fact satisfied by the only othersuggestion for an expression of conditional desirability that I haveseen: that Des(X|Y) = DesXY .4 For this proposal gives us thatDes(T |Y) = DesY 6= DesT , i.e. it is not normalised with re-spect to the desirability measure onT . It should be emphasised,however, that this alternative does not conflict with the principlethat all no-news situations are equal – it also entails, for instance,that Des(Y |Y) = Des(T |Y). It simply does not equate the variousconditional desirability measures onT (it is as if a different scaleis chosen for each conditional desirability measure based on thedesirability of the condition).

To arrive at our candidate expression for conditional desirabilitiesin terms of unconditional ones, we reason as follows. Getting thenews thatXY is true is just the same as getting both the news thatX is true and the news thatY is true. But DesXY is not necessarilyequal to DesX + DesY because of the way in which the desirabili-ties ofX andY might depend on one another. UnlessX andY areprobabilistically independent, for instance, the news thatX is truewill affect the probability and, hence, the desirability ofY . Or itmight affect the desirability ofY directly, because it is the sort ofcondition that makesY less or more desirable. It is natural then tothink of DesXY as equal, not to the sum of the desirabilities ofX

andY , but to the sum of the desirability ofX and the desirability ofY given thatX is true. This gives:

DEFINITION (conditional desirability).If Prob(XY) 6= 0,

Des(X|Y) = DesXY − DesY +DesT .

The DesT term in the definition of conditional desirability is thereto normalise the measure in accordance with the normality condi-tion. Notice that if an agent is sure that some condition is false,then there will be no component of her unconditional attitudes that

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derives from the consideration she gives to that condition’s possi-ble truth. So the conditional desirability measure, like that of con-ditional probability, is not defined relative to any propositions ofprobability zero. Nor is it defined for propositions whose probabilitygiven the truth of the condition is zero. (It is standard in the Jeffreyframework not to define the desirability of news the agent regards asimpossible.)5

Why should degrees of conditional desire be conditional desir-abilities as defined here? Think of the desirability of a propositionas the amount of money an agent is prepared to spend to make ittrue. Obviously not all propositions can be made true in this way,so this treatment is of heuristic value only (though if the traditionof making offerings to the gods are anything to go by, humans areinclined to give it a try for quite a range of them). Our definitionof conditional desirability is now translated as follows: the amountof money you are prepared to spend to makeX true if Y is, is thedifference between the amount you are prepared to spend on makingXY true and the amount you are prepared to spend on makingY

true, i.e. given that you are prepared to pay a certain amount forY ,then it is the extra amount that you are prepared to makeX true aswell.

Suppose that I have the opportunity of buying tickets that maybe exchanged for various items on the menu of a cafeteria and tosimplify let the indifference price of a ticket be the price that is boththe maximum I am prepared to pay for it and the minimum I amprepared to sell it for. Suppose that indifference price of the ticketentitling me to cheese platter accompanied by a glass of red wine is$10 and that of the cheese platter alone is $6. What is the maximumprice I should be prepared to pay for a ticket that entitles me to theglass of red wine if I already have a ticket for the cheese platter (andto get my money back of I don’t)? Surely the difference betweenthe two prices: $4. For if I were prepared to pay $x more, then bycunning arbitrage someone could buy from me for $10 a ticket thatentitled me to both the cheese platter and wine and sell me a ticketat $6 for the cheese platter and a ‘conditional ticket’ at $(4+ x) forthe wine if I have a cheese platter. To their net benefit of $x.

A similar argument would show that if I were prepared to sellmy ‘conditional ticket’ for less than $4, I would also risk being bled

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white. $4 is, thus, the rational price to pay. By analogy, the rationaldegrees of conditional desire to have are those which are related toone’s degrees of unconditional desire by our definition. The analogyis pretty crude, so I don’t consider this to settle the matter. Andwe will do better in later sections when we look at the represen-tation of conditional preferences. For the moment, however, I takeit the various considerations advanced so far give us good reasonto hypothesise that if an agent’s degrees of conditional desire areconsistent with one another and with her degrees of unconditionaldesire then they will go by the proposed definition of conditionaldesirability.

CONDITIONAL PROBABILITIES FROM DESIRABILITIES

I have been arguing that a rational agent’s degrees of conditionaldesire are conditional desirabilities just as her rational degrees ofconditional belief are conditional probabilities. Now it is noteworthythat conditional probability functions obey the probability axioms:for any propositionsX andZ such thatXZ = F , and proposi-tion Y 6= F , Prob(T |Y) = 1, Prob(X|Y) > 0, and Prob((X ∨Z)|Y) = Prob(X|Y) + Prob(Z|Y). So if Prob is a measure of herdegrees of belief, then for anyY of non-zero probability, Prob( |Y)is a probability measure of her degrees of conditional belief givenY . This shows consistency of conditional belief to be like consis-tency of (unconditional) belief. This is as it should be. An agent’sconditional degrees of belief givenX could be the degrees of un-conditional belief of some rational agent who believed thatX. Inparticular, the agent herself could acquire just such degrees of beliefby conditionalising in response to the news thatX was true.

One might expect that the same would hold for conditional desir-abilities. And, indeed, it does as the theorem below shows.

THEOREM 1. If XY = F ,XZ 6= F andYZ 6= F ,

Des(X ∨ Y |Z) == Des(X|Z) · Prob(X|Z)+ Des(Y |Z) · Prob(Y |Z)

Prob(X|Z)+ Prob(Y |Z) .

(Proof in Appendix.)

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COROLLARY 1. If XYZ 6= F andX ∼ YZ 6= F ,

(a) Des(X|Z) = Des(XY |Z) · Prob(Y |XZ) + Des(X ∼ Y |Z) ·Prob(∼ Y |XZ)

(b) Des(X|Z) · Prob(X|Z)+ Des(∼X|Z) · Prob(∼X|Z) = DesT(c) Prob(X|Z) = DesT−Des(∼X|Z)

Des(X|Z)−Des(∼X|Z) .

Proof. (a) Follows directly from Theorem 1 and the fact thatX = (XY ∨ X∼Y ). (b) and (c) follows directly from Theorem 1,and the fact thatT = (X ∨∼X) and Des(T |X) = DesT (normalitycondition). �

Theorem 1 shows that if Des is a desirability measure of an agent’sdegrees of desire, then Des( |Y) is a desirability measure of herdegrees of conditional desire givenY , i.e. that conditional desirabil-ity functions are desirability measures. This may be interpreted asfollows. Assume, as we have claimed, that as a matter of consistencyan agent’s degrees of conditional belief and desire are respectivelyconditional probabilities and conditional desirabilities. Then Theo-rem 1 tells us that consistent degrees of conditional desire relate toconsistent degrees of conditional belief in the same way that con-sistent degrees of (unconditional) desire relate to consistent degreesof (unconditional) belief. This is as it should be. One person’s con-ditional desires may be another person’s desires: those of someonewhose beliefs change in such a way that their new degrees of desireequal their old conditional degrees of desire givenY .6 Then theirnew desires will be consistent only if their old conditional degreesof desire satisfied the desirability axiom.

I will now show that the claim that a rational agent’s degrees ofconditional belief are conditional probabilities can be derived fromthe assumption that an agent’s degrees of conditional desire are con-ditional desirabilities and that conditional desirability functions aredesirability measures of degrees of conditional desire.

THEOREM 2. Let Desy and Proby andDesandProbbe pairs ofdesirability and probability functions such thatDesyX = Des(X|Y)for everyX such thatXY 6= F . Then:

ProbyX = Prob(X|Y) = ProbXY/ProbY.

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(Proof in Appendix)

I take it that Theorem 2 provides a different kind of justification formeasuring degrees of conditional belief by ratios of unconditionalones. For we may interpret it as saying that if an agent’s degrees ofconditional desire are measured by the desirability function, Desy ,which is such that Desy = DesXY−DesY+DesT , then her degreesof conditional belief are measured by the probability function Proby ,which is such that ProbyX = ProbXY/ProbY . This argument shiftsthe burden of justification onto the claim that conditional desirabilityfunctions are the right desirability measures of a rational agent’sconditional desires. It is time to improve on our heuristic argumentsin its favour.

CONDITIONAL PREFERENCES

When I say that if it will be hot tomorrow I prefer that we go tothe beach than to the cinema, I express a conditional preferenceof the simplest variety: a preference for the truth of some propo-sitionX over another propositionY , on the asssumption that someconditionA is true. I will denote such a conditional preference byX >A Y , a notation that will suffice for our purposes since we willnot consider preferences where the assumed condition is varied. Insimilar fashionX ≈A Y will mean that, givenA, the agent is in-different betweenX andY andX >A Y will mean thatX >A Y

or X ≈A Y . In this section I will show that conditional preferencerelations of this simple variety satisfy Bolker’s axioms of preferenceand consequently that they determine the existence of probabilityand desirability functions agreeing with them.Let > be an (unconditional) preference relation that respectsBolker’s axioms and, for any non-contradictory propositionA, let>A be a two-place relation on sets of propositions consistent withA. I propose that an agent’s conditional preferences givenA shouldrelate to her unconditional preferences in accordance with the fol-lowing axiom:

AXIOM OF CONDITIONAL PREFERENCE.Y >A Z iff AY > AZ.

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The axiom of consistent conditional preferences says that you can-not consistently prefer the truth ofAY to that ofAZ and not preferthat, given thatA is the case,Y be true rather thanZ. In both cases(i.e. whetherA is known to be true or not), you are essentiallychoosing betweenY whenA is true andZ whenA is true. Thus,if you prefer getting ice cream on hot days to getting hot chocolateon hot days, then you should prefer that if it is a hot day you getice cream rather than hot chocolate. Your relative attitude to thetwo combinations〈ice-cream, hot day〉 and〈hot chocolate, hot day〉should not depend on whether you know that it is going to be a hotday or not.

THEOREM 3. Let > be a transitive, connected and continuousrelation on a set of propositionsS that respects the axioms of av-eraging and impartiality, and let>A be a two-place relation onSA,the set of propositions inS consistent withA, that respects the axiomof conditional preference. Then>A is a transitive, connected andcontinuous relation onSA that respects the axioms of averaging andimpartiality.

(Proof in Appendix)

Let A be any proposition,�A be a complete atomless algebra ofpropositions minus any propositions inconsistent withA and let>Abe a conditional preference relation defined on�A. Suppose that�Acontains at least one propositionX such thatX >A ∼X. Since byTheorem 3,>A is a preference relation satisfying the assumptionsof Bolker’s Representation Theorem, the following theorem is thenan immediate consequence of the latter.

THEOREM 4 (representation of conditional preferences).(1) Thereexists a desirability measure,DesA, and a probability measure,ProbA,on�A, such that:

X >A Y iff DesAX > DesAY.

(2) If Prob′A andDes′A are another pair of functions which agreewith >A, then there exist real numbers a, b, c and d such that(a)cDesAT + d = 1, (b) ad − bc > 0, and, for all propositionsX in�A, (c) cDesAX + d > 0, and:(d) Prob′AX = ProbAX · (cDesAX + d)

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(e) Des′AX =aDesAX + bcDesAX + d .

It is natural to regard the desirability function, DesA, and probabilityfunction, ProbA, which agree with>A, as measures respectively ofan agent’s degrees of conditional desire and belief in the truth ofpropositions given thatA is the case. Now we have already claimedthat these are measured by conditional desirability and probabilityfunctions. One would expect, thus, that the set of desirability func-tions agreeing with an agent’s conditional preferences should just bethe set of conditional desirability functions generated by the set ofdesirability functions agreeing with her unconditional preferences.Call this the Correspondence Hypothesis.

CORRESPONDENCE HYPOTHESIS. Let> be a preference rela-tion on� and>A the corresponding conditional preference relationon�A. LetK be the set of desirability functions that agree with>,KA the set of desirability functions that agree with>A, andK|A theset of conditional desirability functions, Des( |A), such that Des is amember ofK iff Des( |A) is a member ofK|A. Then:

(i) If Des is a member ofK, then Des( |A) is a member ofKA.(ii) If DesA is a member ofKA, then there exists some member,

Des, ofK, such that DesAX = Des(X|A).

CORRESPONDENCE OF CONDITIONAL ATTITUDES

In this section we establish the conditions under which the Corre-spondence Hypothesis holds true.

THEOREM 5. Let Desbe a desirability function that agrees with>. Then

(a) Des(X|A) > Des(Y |A) iff X >A Y .(b) If Des( |A) is a member ofK|A thenDes( |A) is a member of

KA.Proof. (a) Let Des be a desirability function that agrees with>.

So DesAX > DesAY iff AX > AY . But by the axiom of consistentconditional preference,AX > AY iff X >A Y . So it follows that

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DesAX > DesAY iff X >A Y . But by the definition of condi-tional desirability, DesAX = Des(X|A) + DesA − DesT . Hence,Des(X|A) > Des(Y |A) iff X >A Y .

(b) Let Des( |A) be a member ofK|A. But then from (a),Des(X|A) > Des(Y |A) iff X >A Y . So Des( |A) agrees with>A, and by Theorem 1, Des( |A) is a desirability function. Hence,Des( |A) member ofKA. �

Theorem 5(b) confirms the first part of the Correspondence Hypoth-esis: every conditional desirability function is a desirability functionthat agrees with the agent’s conditional preferences. The second partis more complicated because of the special uniqueness propertiesof probability and desirability representations in Jeffrey’s theory(see Appendices 1 and 2). We show firstly that if some desirabilitymeasure agreeing with an agent’s conditional preferences is a condi-tional desirability function, then every positive linear transformationof it is a conditional desirability function.

THEOREM 6. LetDesbe a member ofK andDesA the member ofKA such that, for all propositionsX in �A, DesAX = Des(X|A).Let Des′A be some other member ofKA such that for some realnumbersa > 0 and b, Des′AX = aDesAX + b. Then there existsa member ofK, Des′, such that, for all propositionsX in�A andYin �:

(i) Des′AX = Des′(X|A)(ii) Des′Y = aDesY + b.

Proof.Let Des and Des′ be members ofS such that for some realnumbersa > 0 andb, Des′Y = aDesY + b, their existence beingguaranteed by Bolker’s Representation Theorem. Then by CorollaryU2.1 (in Appendix 2) it follows that, for allX in �A:

Des′(X|A) = aDes(X|A)+ b.So Des′( |A) is the member, Des′A, of KA such that Des′AX =aDesAX + b. �

It turns out, however, that the second part of the CorrespondenceHypothesis is not true in general: not every desirability functionthat agrees with an agent’s conditional preferences is a conditional

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38 RICHARD BRADLEY

desirability function on the agent’s unconditional preferences. In thefollowing theorem we establish the conditions under which the Cor-respondence Hypothesis is satisfied for zero-normalised desirabilityfunctions. (A zero-normalised function is one assigning zero to thetautology. Since the choice of value for the tautology is arbitrary weconsiderably reduce the amount of tedious calculation by makingthis restriction. The more general case will be handled later on.)

THEOREM 7. Let Des be a zero-normalised member ofK andDesA a zero-normalised member ofKA such that, for all propo-sitionsX in �A, DesAX = Des(X|A). Let Des′A be some otherzero-normalised member ofKA such that for some real numbersb >0 andd, dDesAX + 1> 0 andDes′AX = bDesAX/(dDesAX + 1).Then there exists a zero-normalised member ofK, Des′, such thatDes′A = Des′( |A) iff there exist real numbers,a > 0 and c, suchthat, for all propositionsX in �,

(i) cDesX + 1> 0(ii) Des′X = aDesX/(cDesX + 1)

(iii) bc(cDesA+ 1)− ad = 0(iv) a − adDesA− b(cDesA+ 1) = 0


The question as to whether every zero-normalised member ofKA isa member ofK|A reduces, thus, to the question as to whether wecan be sure of finding, for any choice ofb andd that satisfies theassumptions of the uniqueness theorem, real numbersa andc thatsatisfy conditions (i), (iii) and (iv). Solving fora in (iii) and (iv)above, we get:

1. a = bc(cDesA+ 1)/d

= b(cDesA+ 1)/(1− dDesA)

⇒ 2.c = d/(1− dDesA)

3. d = c/(cDesA+ 1).

So we can always choosea andc in accordance withb andd so as tosatisfy equalities 1 and 2. Then conditions (iii) and (iv) of Theorem

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7 would be satisfied. Furthermore, by substituting forc in 1. we get

5. a = b(dDesA+ 1−DesA)

(1− dDesA)(1− dDesA)= b/(1− dDesA)(1− dDesA).

Then sinceb > 0 and(1− dDesA)(1− dDesA) > 0, it follows thata > 0 as required.

It remains to be seen whether these choices ensure that condition(i) of Theorem 7 holds: thatcDesX + 1 > 0 for all X in �. Nowwe have thatd must be such that for all propositionsX in �A,dDes(X|A) + 1 > 0. So letY andZ be any propositions such thatZ >A T >A Y . Then Des(Z|A) > 0> Des(Y |A) and:

− 1/Des(Z|A) 6 d 6 −1/Des(Y |A)⇒ − 1/(DesAZ − DesA) 6 d 6 −1/(DesAY − DesA)

⇒ 6.1/(DesA− DesAZ) 6 d 6 1/(Des− DesAY).

For very high values of DesAZ and low values of DesAY , d is avery small number. Indeed, in the case of unbounded desirabilityfunctions, this condition forcesd to equal zero. But it is typically as-sumed that desirability functions are not bounded (without boundedfunctions one risks the St Petersburg paradox7).

Suppose then that Des is not bounded and that we make ourchoice ofc on the basis of 3. Then from that fact thatdDes(X|A)+1> 0 it follows that:

(cDes(X|A)/(cDesA+ 1))+ 1> 0

⇒ (cDesAX − cDesA+ cDesA+ 1)/(cDesa + 1) > 0

⇒ (cDesAX + 1)/(cDesA+ 1) > 0

⇒ 7. cDesAX + 1> 0 iff cDesA+ 1> 0.

LEMMA 2. Suppose thatcDes(X|A)+1> 0 and thatY andZ areany propositions such thatZ >A T >A Y . Then if1− dDesA < 0,then(i) A > T andDesAY > DesT , or (ii) A < T andDesAZ <DesT .

Proof.Assume that 1− dDesA < 0. Then:(i) If A > T , thend > 1/DesA. But by 6,d 6 1/(DesA −

DesAY). So 1/DesA < 1/(DesA − DesAY). Suppose DesAY 6

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40 RICHARD BRADLEY

DesT = 0. Then 1/DesA < 1/(DesA−DesAY) iff DesA−DesAYiff DesAY > 0. So DesAY > DesT .

(ii) If A < T thend < 1/DesA. But by 6,d > 1/(DesA −DesAZ). So 1/DesA > 1/(Desa−DesAZ). Suppose that DesAZ >Desa. Then 1/DesA > 1/(DesA − DesAZ) iff DesA < DesA −DesAZ iff DesAZ < 0. �

Suppose that for someA there were at least one propositionY suchthatAY < T and propositionZ such thatAZ > T . Then by Lemma2, 1− dDesA > 0. But by application of 3:

1− dDesA > 0 iff (cDesA+ 1− cDesA)/(cDesA+ 1)

> 0 iff cDesA+ 1> 0.

Then it follows from 7 that, for allX in �A, cDesAX + 1 > 0.However, we require thatcDesX + 1 > 0 for all propositionsX in�, not just those in�A and this is not guaranteed by the assumptionswe have made. For there may be propositionsA, B andC suchthat,B > T > C and such that, for all propositionsX, DesAXis considerably less than DesB and considerably greater than DesC.And then for some choices ofd, we will havecDesB + 1 < 0 andfor other ones,cDesC + 1< 0.

Let s and i be the lower and upper bound of some desirabilityfunction, Des. Then for any real numberc and some propositionA,the fact thatcDesAX + 1 > 0, for all propositionsX in � impliesthat cDesX + 1 > 0, for all propositionsX in �, iff there are noreal numberss ′ < s andi ′ > i such thats ′ > DesAX > i ′. In otherwords, iff the range of values taken by Des for the propositions in�A are fully spread throughout the preference ranking.

DEFINITION. A preference relation,>, over� is said to befullyspreadwith respect to a propositionA iff, for every propositionXin �, there exist propositionsY andZ in �A, such thatAY > X >AZ.

To say that> is fully spread with respect toA is to say that anagent’s preferences for theA-states (the various ways in whichAcan be true) are either unbounded or have the same bounds as thoseon her preferences over all states. There are grounds for thinking thatpreferences ought to be fully spread in this sense. The basic thought

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is that no matter how much more or less desirable on average thetruth of someX is than that of someA, there are alwaysA-stateswhich are more desirable than your averageX one and otherA-states which are less desirable than it. No matter how good (or bad)a state appears at face value, it always seems possible to tell a storyabout a highly unlikely set of circumstances in which it has very bad(or very good) implications.

COROLLARY 7.1. Let> be a preference relation on� that is fullyspread with respect toA and letDesbe any zero-normalised desir-ability function that agrees with>. Let>A be a preference relationon�A andDes′A the zero- normalised member ofKA such that forsome real numbersb > 0 and d, such thatdDes(X|A) + 1 > 0,it is the case that, for allX, Des′AX = bDes(X|A)/(dDes(X|A) + 1). Then there is a zero-normalised member ofK,Des′ such that, for all suchX:

(a)Des′(X|A) = Des′AX

(b)Des′X = bDesX

(1− dDesA)(dDesX + 1− dDesA).


CONCLUSION

If a preference ranking is fully spread with respect to one of itsmembers,A, then Theorem 5 and Corollary 7.1 tells us that theCorrespondence Hypothesis holds true for that member. This meansthat the agent’s unconditional preferences imply the existence ofconditional desirability and probability functions which agree withher conditional preferences givenA, and conversely that her condi-tional preferences givenA imply the existence of unconditional de-sirability and probability functions agreeing with her unconditionalpreferences. We summarise this in the following theorem.

REPRESENTATION THEOREM FOR CONDITIONAL PREFER-ENCES.Let> be a continuous, transitive, and connected relationon � that respects the axioms of averaging and impartiality. Forany non-contradictory propositionA in �, let>A be a preference

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42 RICHARD BRADLEY

relation on�A that respects the axiom of conditional preference.Then:

(1) There exists a desirability measure,Des, and a probability mea-sure,Prob, on�, unique up to the Jeffrey–Bolker transforma-tions, and such thatX >A Y iff Des(X|A) > Des(Y |A); and

(2) If DesA andProbA are a pair of desirability functions agreeingwith >A, then if> is evenly spread with respect toA, thereexists a pair of desirability and probability functionsDes∗ andProb∗ that agree with> and such thatDesAX = Des∗(X|A)andProbAX = Prob∗(X|A).


I take it that this shows that considerations of rational preferenceoblige agents’ degrees of conditional belief and desire to respect thedefinitions of conditional probability and desirability, in the sameway that they oblige the agents’ unconditional beliefs and desiresto respect the probability and desirability axioms. For by (1) if anagent makes his conditional choices on the basis of his degrees ofconditional belief and desires, as derived from his unconditionaldegrees of belief and desire by means of the proposed expressionfor conditional desirability, then his conditional choices will be con-sistent. And by (2), if his conditional choices are consistent thenhis degrees of conditional belief and desire must be related to hisdegrees of unconditional belief and desire in accordance with ourproposed expressions.

APPENDIX 1. REMINDERS OF THEOREMS AND SOME LEMMAS

In Jeffrey’s model an agent’s degrees of belief in and desire for themembers of a set of propositions,S, are measured by a pair of de-sirability and probability functions, Des and Prob, defined onS andsuch that for allX andY in S:

(a) Nonnegativity: ProbX > 0(b) Normality: ProbT = 1(c) Additivity: If XY = F ,

Prob(X ∨ Y) = ProbX + ProbY

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(d) Desirability: If XY = F and Prob(X ∨ Y) 6= 0, then:

Des(X ∨ Y) = DesX · ProbX + DesY · ProbY

ProbX + ProbY

LEMMA 1.

(a)DesXY = DesX · ProbX − DesX∼Y · ProbX∼YProbXY

(b)Des∼X = DesT − DesX · ProbX

Prob∼X(c)ProbX = DesT − Des∼X

DesX − Des∼XProof. (a) By the desirability axiom and from the fact thatX =

(XY ∧X∼Y):DesX · ProbX = DesXY · ProbXY + DesX∼Y · ProbX∼Y

⇒ DesXY = DesX · ProbX − DesX∼Y · ProbX∼YProbXY

(b) and (c). By the desirability axiom and from the fact thatT =(X ∨∼X):

DesT · ProbT = DesX · ProbX + Des∼X · Prob∼X⇒ Des∼X = DesT − DesX · ProbX

Prob∼Xand:

ProbX = DesT − Des∼XDesX − Des∼X. �

Bolker’s representation theorem serves to link an agent’s prefer-ences for the truth of propositions to such probability and desirabil-ity representations. Let� be a complete, atomless Boolean algebraof propositions withF removed and let> be a two-place relationon�, that respects, for allX andY in �:

1. Connexity: X > Y or Y > X2. Transitivity: If X, Y andZ are connected, then ifX > Y andY > Z, thenX > Z

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44 RICHARD BRADLEY

3. Continuity If X1, X2, . . . , Xn is an increasing (or decreasing)sequence of propositions,X the limit of the disjunction (or con-junction) of theXis, and propositionsY andZ such thatY >X > Z, then,Y > Xn > Z for any sufficiently largen.

4. Axiom of Averaging: If XY is a contradiction, then:

X > Y iff X > (X ∨ Y) > Y5. Axiom of Impartiality: Assume thatXY,XZ andYZ are contra-

dictions and thatX ≈ Y . Then if (X ∨ Z) ≈ (Y ∨ Z) for someZ, then(X ∨ Z) ≈ (Y ∨ Z) for all suchZ.

REPRESENTATION THEOREM (Bolker8).(i) Existence: There exists a desirability measure,Des, and prob-

ability measure,Prob, such that, for allX andY in �:

X > Y iff DesX > DesY.

(ii) Uniqueness: Furthermore, ifProb′ and Des′ are another pairof such measures, then there exists real numbers a, b, c and d suchthat for all X in �, (i) cDesT + d = 1, (ii ) ad − bc > 0, (iii )cDesX + d > 0, and:

(iv)Prob′X = ProbX · (cDesX + d)

(v)Des′X = aDesX + bcDesX + d .

COROLLARY U1.1. Let Des and Des′ be such thatDesT =Des′T = 0. Thenb = 0, d = 1, a > 0 and:

Prob′X = ProbX · (cDesX + 1)

Des′X = aDesX

cDesX + 1.

COROLLARY U1.2. Let Des and Des′ be such thatDesT =Des′T = 0 and, for some propositionA, DesA = Des′A. Then:

DesA = (a − 1)/c

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and

Des′X = (cA+ 1) · DesX

cDesX + 1= aDesA · DesX

(a − 1)DesX + DesA.

APPENDIX 2. CONDITIONAL DESIRABILITIES

We extend Jeffrey’s model by introducing conditional desirabilityfunctions defined by:

CONDITIONAL DESIRABILITY. If XY 6= F ,

Des(X|Y) = DesXY − DesY +DesT .

LEMMA 2 (chain rule). If XYZ 6= F ,

Des(XY |Z) = Des(X|YZ)+ Des(Y |Z) −DesT .

Proof.Assume thatXYZ 6= F . Then by the definition of condi-tional desirability,

Des(XY |Z) = DesXYZ −DesZ +DesT

= Des(X|YZ)+ DesYZ − DesZ

since by the definition of conditional desirability DesXYZ =Des(X|YZ) + DesYZ − DesT . Again by this definition, DesYZ= Des(Y |Z) +DesZ −DesT , so:

Des(XY |Z) = Des(X|YZ)+ Des(Y |Z) −DesT . �

The existence of conditional desirability functions follows di-rectly from the existence of the desirability functions in terms ofwhich they are defined. A corresponding uniqueness theorem canbe stated.

UNIQUENESS THEOREM FOR CONDITIONAL DESIRABILI-TIES. Let DesandProbandDes′ andProb′ be any pairs of desir-ability and probability functions defined on� and agreeing with>over�, and letDes( |A), Prob( |A) and Des′( |A), Prob′( |A) be

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46 RICHARD BRADLEY

the corresponding pairs of conditional probability and desirabilityfunctions defined on�A. Then there are real numbers, a, b, c, andd, such that(a) cDesT + d = 1, (b) ad − bc > 0, and, for allpropositionsX, (c) cDesX + d > 0, and such that:

Des′(X|A) = (ad − bc)(DesAX − DesA)

(cDesAX + d)(cDesA+ d) + (aDesT + b)

Prob′(X|A) = Prob(X|A) · (cDesAX + d)(cDesA+ d) .

COROLLARY U2.1. If, for all X, Des′X = aDesX + b then:

Des′(X|A) = aDes(X|A)+ bProb′(X|A) = Prob(X|A).

COROLLARY U2.2. LetDesandDes′ be such thatDesT = Des′T =0. Thenb = 0, d = 1 and:

Des′(X|A) = aDes(X|A)(cDesAX + 1)(cDesA+ 1)

.

COROLLARY U2.3. Let DesandDes′ be members ofK such thatDesT = Des′(T ) = 0 andDes′A = DesA. Then:

Des′(X|A) = Des(X|A)(cDesAX + 1)

.

Proof.Let Des and Des′ be desirability functions that agree withan agent’s preferences. By the definition of conditional desirability:

1. Des′(X|A) = Des′AX − Des′A+ Des′T .

By Jeffrey’s uniqueness theorem, we have that there exists real num-bersa, b, c andd, such that (a)cDesT + d = 1, (b)ad − bc > 0,

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and, for all propositionsX, (c) cDesX + d > 0, and such that:

2. Des′(X|A) == aDesAX + bcDesAX + d −

aDesA+ bcDesA+ d +

aDesT + bcDesT + d

= (aDesAX + b)(cDesA+ d)− (aDesA+ b)(cDesAX + d)(cDesAX + d)(cDesA+ d)

+ aDesT + b= adDesAX + bcDesA− adDesA− bcDesAX

(cDesAX + d)(cDesA+ d) + aDesT + b

= (ad − bc)(DesAX − DesA)

(cDesAX + d)(cDesA+ d) + aDesT + b.

Similarly by the definition of conditional probability:

3. Prob′(X|A) = ProbXA/ProbA

= ProbXA · (cDesXA+ d)ProbA · (cDesA+ d)

= Prob(X|A) · (cDesXA+ d)(cDesA+ d) .

In particular if Des′X = aDesX + b, then:

4. Des′(X|A) = aDesAX + b − aDesA− b + aDesT + b= a(DesAX −DesA+ DesT )+ b= aDes(X|A)+ b

in accordance with Corollary U2.1.Now let DesT = Des′(T ) = 0. Then by Corollary U1.1 of Jeffrey’suniqueness theorem,b = 0 andd = 1. So substituting into 2, wehave:

Des′(X|A) = aDes(X|A)(cDesAX + 1)(cDesA+ 1)

in accordance with Corollary U2.2.

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48 RICHARD BRADLEY

Finally, assume in addition that DesA = Des′A. Then by Corol-lary U1.2,a = cDesA+ 1 and hence:

Des′(X|A) = Des(X|A)(cDesAX + 1)

in accordance with Corollary U2.3. �

APPENDIX 3. PROOFS

Proof of Theorem 1. Assume thatXY = F ,XZ 6= F andYZ 6= F .By the definition of conditional desirability,

(1) Des(X ∨ Y |Z) = Des(XZ ∨ YZ)− DesZ + DesT

⇒ Des(X ∨ Y |Z) · Prob(XZ ∨ YZ)= (Des(XZ ∨ YZ)−DesZ +DesT ) · Prob(XZ ∨ YZ).

But by the desirability axiom:

Des(XZ ∨ YZ) · Prob(XZ ∨ YZ)= DesXZ · ProbXZ + DesYZ · ProbYZ.

So substituting into (1):

Des(X ∨ Y |Z) · Prob(XZ ∨ YZ)= DesXZ · ProbXZ + DesYZ · ProbYZ

+ (DesT − DesZ) · Prob(XZ ∨ YZ).But Prob(XZ ∨ YZ) = ProbXZ + ProbYZ, becauseXY = F . So:

Des(X ∨ Y |Z) · Prob(XZ ∨ YZ)= (DesXZ − DesZ + DesT ) · ProbXZ

+ (DesYZ − DesZ + DesT ) · ProbYZ

= (Des(X|Z) · ProbXZ + Des(Y |Z) · ProbYZ

⇒ Des(X ∨ Y |Z) == Des(X|Z) · ProbXZ + Des(Y |Z) · ProbYZ

ProbXZ + ProbYZ

= Des(X|Z) · Prob(X|Z)+ Des(Y |Z) · Prob(Y |Z)Prob(X|Z)+ Prob(Y |Z)

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by division of the numerator and denominator by ProbZ. �

Proof of Theorem 2. By Lemma 1(c):

ProbyX = DesyT − Desy∼XDesyX − Desy∼X

= Des(T |Y)− Des(∼X|Y)Des(X|Y)− Des(∼X|Y) [by definition of Desy]

= DesT Y−DesY+DesT −Des∼XY+DesY−DesT

DesXY−DesY+DesT −Des∼XY+DesY−DesT[by definition of Des( |Y)]

= DesY −Des∼XYDesXY − Des∼XY .

Now using the desirability axiom we can expand DesY to get:

DesY = DesXY · ProbXY + Des∼XY · Prob∼XYProbY

And substituting into the above:

ProbyX

= DesXY ·ProbXY+Des∼XY ·Prob∼XY−Des∼XY ·ProbY

(DesXY−Des∼XY)·ProbY

= DesXY · ProbXY−Des∼XY · (ProbY−Prob∼XY)ProbY · (DesXY−Des∼XY)

= ProbXY(DesXY−Des∼XY)ProbY · (DesXY − Des∼XY)

= ProbXY

ProbY= Prob(X|Y).9 �

Proof of Theorem 3. (i) If > is connected, then for allX, Y in �A,AX > AY or AY > AX. But then by the axiom of conditionalpreference,X >A Y or Y >A X. So for allX, Y in �A,X >A Y orY >A X. Hence>A is connected.

(ii) If > is transitive then for allX, Y in �A, if AX > AY

andAY > AZ thenAX > AZ. But by the axiom of conditional

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50 RICHARD BRADLEY

preference,AX > AY andAY > AZ iff X >A Y andY >A Z

andAX > AZ iff X >A Z. So for allX, Y in �A, if X >A Y andY >A Z thenX >A Z. Hence>A is transitive.

(iii) If X1, X2, . . . , Xn is an increasing (or decreasing) sequenceof propositions in�A andX the limit of the disjunction (or con-junction) of theXis, thenAX1, AX2, . . . , AXn is an increasing (ordecreasing) sequence of propositions in�A andAX the limit ofthe union (or intersection) of theXis. Furthermore, by the axiomof conditional preference ifT andZ are propositions in�A suchthat Y >A X >A Z thenAY > AX > AZ. If > is continuousandAY > AX > AZ, then it follows that, for all sufficientlylargen,AY > AXn > AZ. But then by the axiom of conditionalpreference, for all sufficiently largen, Y >A Xn >A Z. So>A iscontinuous.

(iv) If > respects the axiom of averaging then for allX andY in�A if XY is a contradiction, then:

AX > AY iff AX > (AX ∨ AY) > AY.

But by the axiom of conditional preference,AX > AY iff X >A YandAX > (AX ∨ AY) > AY iff X >A (X ∨ Y) >A Y . SoX >A (X ∨ Y) >A Y . So>A respects the axiom of averaging.

(v) LetX, Y andZ be members of�A such thatXY ,XZ andYZare contradictions,X ≈A Y and(X ∨ Z) ≈A (Y ∨ Z). ThenAXY ,AXZ andAYZ are contradictions and by the axiom of conditionalpreference,AX ≈ AY and (AX ∨ AZ) ≈ (AY ∨ AZ). If > isimpartial then for all suchZ, (AX ∨ AZ) ≈ (AY ∨ AZ). But thenby the axiom of conditional preference,(X∨Z) ≈A (Y ∨Z) for allsuchZ. So>A respects the axiom of impartiality. �

Proof of Theorem 7. Let Des and Des′ be zero- normalised membersofK. Then by Corollary U1.2 of the uniqueness theorem (Appendix1) there exists real numbers,a andc, such that, for all propositionsX in �,

1. cDesX + 1> 0

2. Des′X = aDesX/(cDesX + 1)

3. a > 0.

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It follows from Theorem 5(b) and the zero-normality of Des andDes′ that Des( |A) and Des′( |A) are members ofKA such thatDes(T |A) = Des′(T |A) = Des(A|A) = Des′(A|A) = 0. Andby Corollary U2.2 of the uniqueness theorem for conditional desir-ability functions (in Appendix 2):

4. Des′(X|A) = aDes(X|A)(cDesAX + 1)(cDesA+ 1)

.

Now let Des′A be any other zero-normalised member ofKA. Then itfollows from Theorem 4 that there exists real numbers,b > 0 andd, such that for all propositionsX in �A,

5. dDes(X|A)+ 1> 0

and:

6. Des′AX = bDes(X|A)/(dDes(X|A)+ 1).

It follows that Des′( |A) = Des′A iff the real numbers,a andc, aresuch that, for all propositionsX:

bDes(X|A)/(dDes(X|A)+ 1)

= a(Des(X|A)/((cDesAX + 1)(cDesA+ 1))

⇒ b(cDesAX + 1)(cDesA+ 1)

= a(dDes(X|A)+ 1)

⇒ bc2DesAX · DesA+ bcDesAX + bcDesA+ b= adDesAX − adDesA+ a

⇒ 7.DesAX(bc2DesA+ bc − ad)= a − adDesA− b − bcDesA.

Identity 7 must hold for any propositionX in�A. But by assumption�A contains at least one propositionX >A∼X. Then by the axiomof conditional preferenceAX > A∼X and so DesAX > DesA∼X.It then follows that:

8. DesAX(bc2DesA+ bc − ad)= DesA∼X(bc2DesA+ bc − ad).

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52 RICHARD BRADLEY

But since DesAX 6= DesA∼X, it follows that:

9. bc2DesA+ bc − ad = 0

and hence from 7, that:

10. a − adDesA− b(cDesA+ 1) = 0.

Hence Des′A is the member, Des′( |A), of K|A iff there exist realnumbersa andc, that satisfy 1, 2, 3, 9, and 10. �

Proof of Corollary 7.1.We set:

1. c = d/(1− dDesA);2. a = b(cDesA+ 1)/(1− dDesA)

= b/(1− dDesA)2.

So:

3. d = c/(cDesA+ 1).

As before it follows thata > 0.Then from the fact thatdDes(X|A)+1> 0, it follows as before,

by substitution of 3 ford, that:

cDes(X|A)+ cDesA+ 1

cDesA+ 1)> 0.

But by definitioncDes(A|X) = cDesAX − cDesA. So:

4.cDesAX + 1

(cDesA+ 1)> 0.

Since the preference relation> is fully spread with respect toA, itfollows that, for all propositionsX in�, there exists propositionsYandZ, such that DesAY > DesX > DesAZ. So in particular thereexist propositionsY andZ, such that DesAY > DesT > DesAZ.So by Lemma 2, 1−dDesA > 0. Hence by 3,cDesA+1> 0. Thenby 4, cDesAX + 1 > 0. But since the preference relation is fullyspread with respect toA, the fact thatcDesAX + 1> 0 for allX in�, implies thatcDesX+1> 0 for all suchX. It follows that for any

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value ofb andd, if we choose our values fora andc in accordancewith 1 and 2, then we will satisfy the conditions (i), (iii) and (iv) inTheorem 7.

Finally by Bolker’s uniqueness theorem we know that there issome member ofK, Des′, such that, for all propositionsX in �:

5. Des′X = aDesX/(cDesX + 1).

So we satisfy condition (ii) of Theorem 7. Since all the conditionsof Theorem 7 are now satisfied, it follows by application of thetheorem that the function Des′ is the member ofK such that, forall propositionsX in �A:

Des′(X|A) = Des′AX.

And by substitution of 1 and 2 fora andc in 5.

Des′X = bDesX

(1− dDesA)(dDesX + 1− dDesA). �

Proof of Representation Theorem. (1) by Bolker’s RepresentationTheorems there exists a pair of desirability and probability function,Des and Prob, unique up to the Jeffrey–Bolker transformations, thatagree with>. And by Theorem 5, for any propositionA, the pairof conditional probability and desirability functions, Prob( |A) andDes( |A), are probability and desirability measures that agree with>A.

(2) Let DesA and ProbA be any pair of desirability and probabilityfunctions agreeing with>A. Then by Theorem 4, there exists a pairof zero-normalised desirability and probability functions, Des′

A andProb′A that agree with>A, such that Des′AX = DesAX − DesATand Prob′AX = ProbAX.

Similarly by Theorem 4, there exists a pair of zero-normaliseddesirability and probability functions, Des′′A and Prob′′A that agreewith >A, and such that Des′′AX = Des(X|A) − Des(T |A) andProb′′AX = Prob(X|A). It then follows by Theorem 6, that there ex-ists a zero-normalised desirability function Des′′, that agrees with>,and such that Des′′X = DesX−DesT and Des′′(X|A) = Des′′A(X).

Since Des′′A and Des′A are both zero- normalised desirability func-tions in agreement with>A, and Des′′ a zero-normalised desirabil-ity function Des′ that agrees with>, and such that Des′(X|A) =

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54 RICHARD BRADLEY

Des′A(X). And by Theorem 5 and the fact that DesAX = Des′AX +DesAT , it then follows by Theorem 6 that there exists a desirabilityfunction Des∗ that agrees with> and such that Des∗X = Des′X +DesAT and Des∗(X|A) = DesAX.

Finally, since DesA and Des∗ are desirability functions such thatDes∗( |A) = DesA, it follows from Theorem 2 that, for all proposi-tionsX in �A, ProbAX = Prob∗(X|A). �

NOTES

1. See Stalnaker (1984), Ch. 6, for a clear statement of the position and thearguments in its favour.

2. See in particular Jeffrey (1983).3. See, for instance, Earman (1992) for a more detailed demonstration of this.4. Jeffrey suggests it in an exercise (Exercise 13 of Chapter 5 of Jeffrey, 1983)

and Sobel (1989) endorses it for evidentialist decision theory.5. I am inclined to think that a degree of belief of zero in any contingent propo-

sition is unjustified, but it may be argued that real agents do have full beliefsin some non-tautological propositions. I am not sure how it is supposed thatone could determine that this is the case, as opposed to the agent having avery high degree of belief in them. Certainly behavioural evidence will be tooopen to reinterpretation to be conclusive. But in any case, if countenanced,conditional degrees of desire given someY of probability zero, would haveto be treated specially, since the desirabilities of propositions with probabilityzero are not defined in Jeffrey’s framework.

6. I do not claim that they should, only that they can.7. See Jeffrey (1983).8. Proof in Bolker (1966).9. Jeffrey (1983) offers a similar derivation based on the assumption that DesyX =

DesXY (solution to exercise 13, p. 93). As we noted before, such an assump-tion entails that DesyT = DesY and hence that the two functions are scaleddifferently.

REFERENCES

Bolker, E. (1966), Functions resembling quotients of measures,Transactions ofthe American Mathematical Society124: 292–312.

Earman, J. (1992),Bayes or Bust? A Critical Examination of Bayesian Confirma-tion Theory, Cambridge, MA: MIT Press.

Jeffrey, R.C. (1983),The Logic of Decision, Chicago: University of ChicagoPress.

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Sobel, J.H. (1989), Partition theorems for causal decision theorists,Philosophy ofScience56(1): 70–94.

Stalnaker, R.C. (1987),Inquiry, Cambridge, MA: Bradford Books.Teller, P. (1973), Conditionalization and observation,Philosophy of Science26:

218–258.

Address for correspondence:Richard Bradley, Centre de Recherche en Episté-mologie Appliquée, 47 rue des Vignes, F-75016 Paris, FrancePhone: +33 142881182; E-mail: [email protected]

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