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THE DUTCH BOOK ARGUMENT: ITS L O G I C A L FLAWS, ITS S U B J E C T I V E S O U R C E S *

(Received 16 May, 1978) Is there a plausible way of interpreting probability statements, which enables us to understand why (ve should accept the axioms from which probability theory is typically developed? Consider the subjectivist's (or personalist's) interpretation. The subjectivist takes probability statements to be statements about a person's degrees of belief and views the probability axioms as having normative import for these. The justification for the axioms, so viewed, is to be found in various arguments for the claim that a person's degrees of belief as a whole must conform to the axioms (i.e., must be 'admissible') if the person is to be considered rational. 1 One of these arguments is the Dutch Book Argument (the DBA), our present subject. A recent version of the DBA put forward by Frank Jackson and Robert Pargetter attempts to meet certain objection to previous versions by appealing to a universalizability principle. 2 In this paper we subject the Jackson-Pargetter DBA to a critical analysis and conclude that, for many reasons, it fails to sustain the subjectivist's claim. The DBA proceeds by way of the well-known betting quotient method of measuring degrees of belief. Some definitions are needed to see how this is done .3 Q is X's forced betting quotient (FBQ) relative to proposition p in situation S if in S X is forced to select a betting quotient for p in ignorance of whether he will be betting for or against p at that quotient, and the quotient X selects for p is Q. Let a function f from a field ~,~ of propositions to [0, 1] be said to be a probability measure just in case it satisfies the following axioms: (i) (ii)

I f p is in ~- and p is logically necessary, then f ( p ) -- 1 ; If p and q are in # - and p and q are lo#cally incompatible, then f ( p v q) = f ( p ) + f(q).4

Philosophical Studies 36 (1979) i9-33. 0031-8116/79/0361-0019501.50 Copyright 9 1979 by D. Reidel Publishing Co., Dordrecht, Holland, and Boston, U.S.A.

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If g is a function from a set S of propositions to [0, 1 ], then we shall call g admissible just in case there is a probability measure f o y e r the closure of S under negation and disjunction such that the restriction of f to S is g.S We may think of a person's betting quotients relative to a given set of propositions as determined by a function from that set to [0, 1]. We shall call that function his betting function and say that his set of betting quotients is admissible if, and only if, his betting function is. We shall say that X and Y are in a competitive betting situation (a CBS), with X the 'quotient-maker', just in case X is forced to select betting quotients for each member of some set {Pl, ..., Pn }of propositions under the following conditions: (i) (ii) (iii) (iv) (v) (vi)

The size and direction of X's bet regarding each Pi are determined by X, who knows X's betting quotient, Q/, for each Pi; X specifies these quotients in ignorance of the directions and sizes of his bets: Y wants X to lose, and will try to bring it about that X does lose; X and Y both know the Dutch Book Theorem (explained below); Y is very good at spotting inadmissibility in betting quotients; X realizes that (i) - (v) are true.

We can now state quite simply a subjectivistic criterion for determining person's degree of belief (DB) in a proposition: X's degree of belief in a proposition p is Q if, and only if, Q is the betting quotient X would assign to p in a CBS. 6 If we think of a person's degrees of belief relative to a given set of propositions as determined by a function from that set to [0, 1], we may call that function his belief function and say that his set of degrees of belief is admissible if, and only if, his belief function is. Suppose X selects betting quotients for each member of a set of propositions and that Y then chooses, for each propositoon, the direction of X's bet and the stake. If it is theoretically possible for Y to make his choices such that, whatever the outcome, X will sustain a net loss, then X's set of quotients is said to be incoherent. 7 If Y takes advantage of X's incoherent set of quotients in this way, then Y is said to have made a Dutch Book (or simply to have made book) against X. The DBA is based upon a mathematical theorem, the Dutch Book Theorem (DBT), which for our purposes can be regarded as saying: a set of

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betting quotients is coherent if and only if it is admissible. That is, i f X selects quotients and Y then chooses stakes and bet directions, then: it is theoretically possible for Y to make book against X if, and only if, X's set of quotients is inadmissible.S The underlying idea of the DBA is taken by Jackson and Pargetter to be "that a course of action must lead to net loss is irrational". (p. 404). Now it seems rather plausible that, given this idea, we should hold the choosing of inadmissible betting quotients in a CBS to be irrational. By the DBT, if X makes such a choice then it is theoretically possible that Y should arrange things so that X will be bound to sustain a net loss. And since we are imagining that X and Y are in a CBS, it seems that this theoretical possibility will be realized. So perhaps it is irrational to choose inadmissible betting quotients in a CBS. But how do we get from here to the desired conclusion of the DBA that, in every situation, it is irrational to have an inadmissible set of degrees o f belief? Jackson and Pargetter suggest the following strategy (p. 405). First, since the FBQ's chosen in a CBS are degrees of belief ("they are determined by degrees of belief"), we may deduce from the fact that it is irrational to select an inadmissible set of quotients in a CBS that it is also irrational to have an inadmissible set of degrees of belief in a CBS. Then they propose to justify dropping the qualification 'in a CBS' by appeal to what they call the principle of universalizability. For reference, we shall explicitly mark off the main stages of this argument. Stage I attempts to show that: (1)

It is irrational to select an inadmissible set of FBQ's in a CBS.

Stage II is the inference from (1) and the premise "the FBQ's chosen in a CBS are degrees of belief' to: (2)

It is irrational to have an inadmissible set of degrees of belief in a CBS.

In Stage III, the universalizability principle is used to conclude that: (3)

It is irrational to have an inadmissible set of degrees of belief in a normal situation. 9

Let is examine in detail each of these stages.

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I

Stage I is the location of one serious infirmity. (1) is, apparently, thought to be a consequence of the DBT taken together with what Jackson and Pargetter call "the essential insight about rationality that underlies the DBA", viz., that "a course of action that must lead to net loss is irrational". Evidently, Jackson and Pargetter have assumed that a choice by X o f inadmissible FBQ's in a CBS must (given the DBT) lead to net loss. But this assumption, though initially plausible, is false. There is no guarantee that Y will make book in a CBS even if it is theoretically possible to do so. That X's FBQ's are inadmissible may, for example, be essentially undiscoverable for Y, due to his not knowing some requisite logical fact - a fact known, perhaps, to no one at all, or known only to a handful of expert logicians. Or it could be that, although Y knows all the requisite facts, he is nevertheless not able to detect the inadmissibility, notwithstanding his enormous talents in this direction. But even if it were true that choosing inadmissible FBQ's in a CBS always and necessarily led to net loss, (1) would still be open to doubt, due to certain problems regarding the so-called "essential insight about rationality underlying the DBA". Need we accept the claim that "a course of action that must lead to net loss is irrational"? These seem to be several reasons not to do so. But before we consider them, let us mention one possible objection which we shall not be making. One might argue that it is not obviously irrational to care very little for money. Therefore, since it is a loss o f money that's at issue, the 'essential insight' is a falsehood. Clearly this is a sound objection as far as it goes; but it may be replied by some defenders of the DBA that money is just standing in for whatever may truly be considered to be, as Ramsey puts it, 'goods', or "things a person ultimately desires"a~ We are not convinced that such a reply can be made without running into difficulties or undermining one of the principal reasons for preferring the betting behavior approach to the axiomatic decision-theoretic method. 11 However, to avoid getting involved in the complexities of utility theory, we shall not pursue this line o f thought. There are more serious problems with the 'essential insight'. A course of action may be bound to lead to a net loss of goods and yet be such that one could reasonably fail to see this. We shall mention j.ust one o f several examples of this type which are relevant to the DBA. Suppose the better is poor at performing computations of the sort needed to detect in-

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admissibility but pretty good on the whole at detecting inadmissibility intuitively. From past experience he knows that he's more likely to detect inadmissibility if he just goes by the way he feels about the set of FBQ's than he would be if he were to attempt the necessary calculations. Surely this would be a rational policy, and surely it would be rational for him to follow this policy even on those exceptional occasions when his 'nose' does lead him astray. But on any such occasion, he would be rationally pursuing a course of action that would be bound to lead to net loss. Perhaps this objection depends on taking the Jackson-Pargetter formulation of the 'essential insight about rationality' too literally. It may be thought that the principle upon which the argument is intended to rely is the following, weaker, claim: If a course of action is known by X to lead necessarily to net loss, then it is irrational for X to pursue that course o f action. This would indeed be a more plausible foundation for the DBA, but there are problems. First, this principle is not strong enough to do the work Jackson and Pargetter require. At any rate, it is not sufficient support for conclusion (1), even given the dubious assumption that choosing inadmissible FBQ's in a CBS must lead to net loss. It would seem that they could proceed from the weaker principle only by weakening the conclusion of their whole argument to read: In a normal situation, it is irrational to have a set of degrees of belief that one knows to be inadmissible. But secondly, although the weaker principle is more plausible than the stronger one, it too seems to be false, since a course of action known to lead to net loss may be preferable to any available alternative. The following two examples should make this evident. Suppose that X (as the quotient-maker) is in a CBS vis-a-vis Y. In the first example, we suppose that X is betting on the outcome of next week's football games, and that X knows almost nothing about football. Given the assumption that there is some limit to the size of the stakes Y may choose, 12 it may well be rational for X to choose inadmissible FBQ's in the hope that Y will try to make book rather than bet the individual games. It may be X's reasonable assumption that, although he is bound to lose money in either case, he would lose less in the former than in the lat~er) 3 In the second example X and Y are famous logicians. Y has

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proved, but not published, the result that ~b and ~b are inconsistent. X has discovered this result also, but by the unacceptable method of surreptitiously rifling through Y's files. Since X and Y have rather different specialties, it is X's reasonable assumption that if he chooses FBQ's for q~and ~ that add up to his FBQ for (4 v q~), all of these propositions being among those concerning which he must bet, then Y will become confirmed in his suspicions o f X, and X ' s reputation will be lost. Since X would rather lose some money than his reputation, he chooses inadmissible FBQ's. A final difficulty concerning the 'essential insight' is that, even in its strong form - i.e., as the principle that a course o f action bound to lead to net loss is irrational, whether or not it is known to lead to net loss - it is insufficient suplSort for conclusion (1), even given the assumption that choosing inadmissible FBQ's in a CBS is bound to lead to net loss. Why is this? Presumably, we are to read 'net loss' as meaning net loss in the long run, or something of the sort. Otherwise, the 'essential insight' won't be plausible at all, since a course of action that involves net losses in the beginning may lead to large net gains if pursued long enough. But if we read the 'essential insight' this way, it clearly does not help to support conclusion (1). Setting inadmissible FBQ's on a particular occasion may be part of a long-range strategy that is likely to lead to an ultimate net gain. Suppose, for example, that the propositions to be bet on concern poker hands (i.e. assume the participants are playing a kind of poker) and that the better is trying to 'sucker' his opponent into participating in a long series of bets by getting him to believe that he (the better) is very naive. Thus, he may purposely set inadmissible FBQ's in the beginning in the reasonable belief that, after a few losses, he would be able to 'make a killing' 14. Thus, the reasons offered for accepting conclusion (1) are weak, and the argument seems to fail at the very first stage. Let us now turn to

STAGE

II

In this section, we assume (1) for the sake o f argument, i.e. we assume that it is irrational to select an inadmissible (and therefore incoherent) set of FBQ's in a CBS. We then analyze the inference to (2). As we said earlier, Jackson and Pargetter make use of this premise: The FBQ's set by X in a CBS are X's degrees of belief.

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But what does it mean to say FBQ's are degrees of belief? Are they suggesting that a FBQ is the very same entity, thing or object as a degree of belief? So that anything one says about the former is said about the latter? If so, the premise is surely dubious. After all, one sets or specifies FBQ's usually at some particular moment and usually in the context of setting up a bet or series of bets; whereas one doesn't usually set or specify one's degrees o f belief - indeed, it is not clear what it even means to 'set one's degrees of belief'. Perhaps it would mean something like 'specifying what one's degrees of belief are'. Again, one might say of X's degrees of belief that they are reflected in the FBQ's set by X in a CBS; but this would surely not mean that the FBQ's set by X in the CBS are reflected by the FBQ's set by X in the CBS. Furthermore, it seems obvious that in many specific CBS's, it would be reasonable to wonder whether X (the quotient-maker) ought to set FBQ's that match his degrees of belief, i.e. whether, for every proposition p to be bet on, X's FBQ for p should be r if and only if X's DB in p=r. Indeed, according to one plausible decision-theoretic analysis (discussed below), there are many conceivable CBS's in which it would be irrational for X to set FBQ's that match his DB's. Such a view would, of course, be absurd if a person's FBQ for proposition p is the very same thing as his DB in p. What then is a more plausible view of the relation between FBQ's and DB's? Consider what de Finetti has written about finding an operational 'definition' (or 'device' as he sometimes calls it ['Foresight', p. 102]) of 'probability' or degrees of belief: The criterion, the operative part of the definition which enables us to measure it, consists in this case of testing, through the decisions of an individual (which are observable), his opinions (previsions, probabilities), which are n o t directly observable. (Theory of Probability, Vol. I, p. 76).

It can be seen that de Finetti holds, as do most subjectivists, that a person's DB's are not directly observable; however since these states of mind can affect the agent's behavior, it is suggested that we use the person's decisions (the expressions of which are observable) to measure these degrees of belief. The betting quotient method then provides us with one such 'device'. The rationale for adopting such a measure need not be gone into here, since it has been described in detail elsewhere, is However, it should be emphasized that using FBQ's is only one of a variety of methods for measuring a person's degrees of belief. Leonard Savage, for example, advocates asking the person a series of questions as to "what he would do in such and such a situation" to determine his probabilities, and as Savage notes "Several schemes of behavioral, as opposed to direct, interrogation have been proposed ''16.

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Furthermore, none of these operational devices is thought to yield perfectly accurate results in all situations. Thus, de Finetti writes, "Of course, a device is always imperfect, and we must be content with an idealization" 17 What the above account suggests is that 'degree of belief' is a theoretical term standing for 'states of mind' which can be measured, at least in principle, more or less accurately in a variety of ways and in a variety of different situations. Thus, it seems possible that the Jackson-Pargetter premise merely expresses something like: (*)

If X is in a CBS setting FBQ's, then X will set a FBQ for proposition p = r i f f X ' s DB in p = r.

The most plausible chain of reasoning we have been able to construct according to which (2) is inferred from (1) and something like (*) goes as follows: (a-l) (a-2) (a-3) (a-4)

If X is in a CBS at time t and X is disposed to choose inadmissible FBQ's at t, then X is disposed to act irrationally at t. If X is in CBS at t, then X is disposed to choose inadmissible FBQ's at t iff X has an inadmissble set of degrees of belief at t. Therefore, if X is in a CBS at t and X has an inadmissible set of degrees of belief at t, then X is disposed to act irrationally at t. Therefore, if X is in a CBS at t, then it is irrational for X to have an inadmissible set of degrees of belief at t.

Now (a-l) seems to follow from (1), and (a-2) can be regarded as a vaguer but more plausible version of the idea expressed by (,).18 (a-3) follows directly from (a-l) and (a-2). And (a-4), which is just a rewording of (2), might well be thought to follow from (a-3). But however Jackson and Pargetter may have reasoned in inferring (2) from (1), we believe that there are reasons for questioning the inference. To some extent, our doubts are due to the conviction that the consideration relevant to judging the rationality of setting particular FBQ's are different from those relevant to judging the rationality of having particular degrees of belief. Consider the following example. S is in a CBS and he is asked to set FBQ's for the propositions p and - p . Now suppose S knows that either p is a logically necessary proposition or - p is (we may suppose that S knows that one or the other proposition was proven recently by a prominent logician). Unfortunately, he does not know which proposition

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is the logically necessary one. We can imagine that he believes it as likely that p is true as that - p is, so S's degree of belief in p = S's degree of belief in - p = 1/2, Now S may know the DBT and he may very well know that to set an admissible system of FBQ's, he must specify either 0 or 1 for p: anything else would leave him open to a Dutch Book. Since we have granted for the sake of argument that it is always irrational (in a CBS) to set FBQ's that one knows to be inadmissible, we can conclude that it would be irrational for S to set his FBQ f o r p to be 1/2. However, it is by no means obvious that it would be irrational for S to have a degree of belief in p -- i/2. On the contrary, since S has no reason for being more confident of p than - p or of - p than p, it would seem that he is being reasonable in not having a degree of belief in p of 0 or 1. Examples such as this have convinced us that whatever reasons we may have for judging that it is irrational to set an inadmissible system of FBQ's in a CBS, these are not necessarily reasons for concluding that it is irrational to have an inadmissible system of degrees of belief in a CBS. The general point made above has recently been given additional support by an analysis of Ernest Adams and Roger Rosenkrantz which consists, ironically, in applying a subjectivistic decision model to CBS's, along the lines traced out by Leonard Savage and Richard JeffreyJ 9 Thus, a person X in a CBS (as the quotient-maker) can be regarded as being in a decisiontheoretic situation in which he must decide what FBQ's he should set (so the possible acts become identified with the possible FBQ's). When the situation is so analyzed, it is by no means obvious that X should always set FBQ's that match his DB's. Indeed, a result of the Adams-Rosenkrantz analysis is that in many CBS's, X would do well to set FBQ's that do not match his DB's if he wishes to maximize expected utility. Hence, this analysis also leads to the conclusion that reasons for setting FBQ's for the propositions P l , P 2 , ..., P n . = r l , r~, . . , rn respectively are not necessarily reasons for having DB's in Pa, P2, respectively. Thus, we conclude that Stage II of the argument also fails: we simply have not been given compelling reasons for concluding (2) from (1). 9. . , P n = r l , r2 . . . . , r n ,

STAGE

II1

The final stage of this DBA involves the use of the universalizability principle, which Jackson and Pargetter regard as their main contribution to the discus-

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sion of the argument. The inference of (3) from (2) by means of this principle is a most important step in the argument, since (2) alone would give us no reason at all to think that having inadmissible degrees of belief is, in general, irrational. But is the inference valid? Let us examine the Jackson and Pargetter discussion of universalizability. They think of their principle as an analogue of an ethical principle which bears the same name and which they formulate as follows: "If act A is (morally) right and B is like A in all relevant respects, then B is right". (p. 404). In other words, it can't be morally right for me to perform a certain kind of act but wrong for you to do so (as long as the two individual acts are alike 'in all relevant respects'). The epistemological analogue of this, advocated by Jackson and Pargetter, is: "If it is rational to believe proposition p in situation S1, and situation $2 is, in all respects relevant to the truth of p, like S1, then it is rational to believe p in $2". (p. 404). Modified for degrees of belief, it reads: "If it is rational to believe p in S1 to degree R and $2 is identical to S1 in all respects relevant to the truth of p, then it is rational to believe p in $2 to degree R". (pp. 404-5). A formulation which is perhaps better for the purposes of deriving (3) from (2), and which seems no less plausible than this one, may be obtained by replacing the word 'rational' by 'irrational'. What may be said in favor of this principle (hereafter, the UP)? Unfortunately, we are on our own here, since Jackson and Pargetter simply "take this principle to be an evident truth - one such that the best argument for it is to state it". (p. 404). One reason it is difficult to assess the UP is that the words 'relevant to the truth of p ' are ambiguous - and doubly so. First, it is not clear whether the relevance in question is to be taken as epistemological or causal. 2~ But it is clear that if 'relevant' is read as 'causally relevant' the UP is false. Let us illustrate. Suppose John Doe is locked in an escape-proof, sound-proof, lightproof, communication-proof, box with a tamper-proof twenty megaton timebomb set to go off in five minutes. We can imagine two possible situations, S1 and $2. In S1 there is a sign on one of the interior walls o f the box which explains about the bomb. In $2 there is no sign. For all John Doe knows the bomb is somebody's lunchbox. Let p be the proposition that John Doe will be dead in ten minutes. $1 en $2 are clearly identical in all respects causally relevant to the truth of p." But one could hardly maintain that John Doe's believing p would have to be equally rational in the two situations.

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Hence we may as well read 'relevant' in the UP as 'epistemologically relevant'. For the purpose of illustrating the remaining ambiguity let us ask you to imagine that the sign which explains about the bomb is hanging on an exterior wall of the box rather than on an interior wall. Are $1 and $2 alike in all respects epistemologically relevant to the truth of p? For John Doe, yes. But for Joe Smith, who is walking around outside, trying to figure out what a box like this is doing in his backyard, no. (Suppose, in this example, that the sign - present in S1 and absent in $2 - includes the information that John Doe is in the box.) Thus, any question concerning whether the two situations are alike in all respects epistemologically relevant to the truth of some proposition is essentially incomplete - and the possibility of understanding the question as completed in different ways gives rise to ambiguity. We can now state a version of the UP free from both of these sorts of ambiguity: [ rational/. If it is ]irrational~m situation S1 for person X to believe proposition p (to degree r), and situation $2 is like situation SI in all respects epistemologically relevant for X to rationall the trutil of p, then it is irrational/ in situation $2 for person X to believe proposition p (to degree r).

In this disambiguated form the LIP certainly seems plausible. And the auxiliary assumption - that for every person X, every proposition p, and every normal situation SN, there is a competitive betting situation which is just like SN in all respects epistemologically relevant for X to the truth of p - may also be acceptable. [Note: There is room for doubt here, though. It could be maintained that the fact that one is in a CBS is always epistemologically relevant.] Does the inference, then, from (2) to (3) go through? The inference from (2) to (3) depends on the truth of the disambiguated UP. But we shall argue that if this form of the UP is true, the very foundations of the DBA are destroyed. It is trivially true that factors which are epistemologically relevant to the truth of a given proposition are relevant to the rationality of belief in that proposition. The fact that Baby Bear finds no porridge in his bowl upon his return is epistemologically relevant to, and can make rational his belief in, the proposition 'somebody's been eating my porridge'. What is more controversial is the thesis: (R)

Factors which are not epistemologically relevant to the truth of a proposition may nevertheless be relevant to the rationality, or irrationality, of belief in that proposition.

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To see why (R) might be thought acceptable, consider this example; during some future Inquisition, the penalty for having heterodox beliefs is torture to death, and techniques are then available for ascertaining with certainty just what a person's beliefs are, These facts will not be relevant to the truth of orthodox beliefs but - some would maintain - they would be relevant to the rationality of having orthodox beliefs. (cf. Pascal's wager). Whether or not (R) is true, it does seem to be presupposed by (2) of the DBA. The factors that are supposed to make it irrational to have an inadmissible set of beliefs in a CBS are entirely irrelevant, epistemologically, to the truth of the propositions in question. The fact (if it is a fact) that one will be bound to lose money unless one's degrees of belief are admissible just isn't epistemologically relevant to the truth of those beliefs. Hence if (R) is false, so is (2) of the DBA. Therefore, (R) must be true if the DBA is to be accepted. Suppose, then, that (R) is true. This turns out to be just as bad for the DBA, because (R) can be true only if the UP is false. Clearly, two situations could be identical in all respects epistemologically relevant to the truth of some proposition and yet differ in some respects relevant to how prudent belief in the proposition might be. (Compare our present situation with that of people at the time of the imagined future Inquisition.) But if the UP is true, it cannot be rational in the one situation, and irrational in the other, to believe the proposition. On the other hand, (R) implies that it might well be rational to believe the proposition in one o f two such situations, and irrational in the other. The Jackson-Pargetter DBA seems, thus, to founder on a dilemma. (R) is either true or false. But if (R) is true, then the UP is false, and (2) cannot support the conclusion of the DBA. And if (R) is false, we have no reason to accept the essential lemma (2).

CONCLUSION

In closing we should note that the DBA is only one of several types of argument which have been offered in support of the contention that our beliefs ought to conform to the axioms of probability theory. 21 We certainly do not wish to cast doubt on all such arguments. But we think it safe to conclude here that the DBA, at least in the form in which we have been considering it, is unequal to its appointed task. It should also be said that, although we have criticized in detail one

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particular version o f the DBA, m a n y o f our o b j e c t i o n s apply to o t h e r versions as well. I n d e e d ; we believe t h a t every DBA we have e x a m i n e d is subject to at least s o m e o f o u r o b j e c t i o n s .

Wake Forest University University of California, Berkeley

NOTES * Some of the material in this paper was presented in a seminar on the foundations of subjective probability theory given by Ernest Adams and the Berkeley author in the fall term in 1977. We are grateful to those who participated in the seminar, in particular to Professor Adams and Dr. Roger Rosenkrantz, for ideas and criticisms. We would also like to thank John Vickers for his comments on this paper. Earlier versions of the paper were delivered at the Wake Forest University Physics Seminar (January 19, 1978) and at the University of California Logic Colloquim (February 24, 1978). Research for this paper was supported by a University of California Research grant. The foremost exponent of this sort of approach to probability theory is Bruno de Finetti, who writes in 'Foresight: Its logical laws, its subjective sources', in: Studies in Subjective Probability, ed. by H. E. Kyburg Jr. and H. E. Smokler (Wiley, New York, 1964), p. 103, that the whole calculus of probability may be regarded as "a set of rules to which the subjective evaluation of probability of various events by the same individual ought to conform if there is not to be a fundamental contradiction among them". It should be mentioned that not all who take this general approach to probability theory would insist on the strong rationality claim that is the conclusion of the version of the Dutch Book Argument we consider here. For example, in de Finetti's more recent work, Theory of Probability: A Critical Introductory Treatment, Vol. I, p. 95, it is claimed that the conditions of admissibility "must exclude the possibility of certain consequences whose unacceptability appears expressible and recognizable to everyone .... Let this be said in order to make clear that such conditions, although normative, are not (as some critics seem to think) unjustified impositions of a criterion which their promoters consider 'reasonable': they merely assert that 'you must avoid this if you do not want...' (and there follows the specification of something which is obviously undesirable)." 'A modified dutch book argument', Philosophical Studies 29 (1976), pp. 403-407. All pages references will be to this article unless otherwise indicated. 3 The definitions of 'forced betting quotient' and 'competitive betting situation' we give here are, with modifications, those of Jackson and Pargetter, ibid., pp. 404/5. The most significant changes we have made are in the definition of 'competitive betting situation'; these resulted from the need to avoid certain obvious objections to the DBA. None of our objections to the argument depends on these changes. The definitions of 'bet' and 'betting quotient' are presupposed in our discussion (as they were in the Jackson and Pargetter paper): for these definitions, the reader can consult any standard work on the topic such as Brian Skyrms' Choice and Change: An Introduction to Inductive Logic, 2nd ed., (Dickenson, Encino, 1975). 4 This characterization of probability measures is adapted from A.N. Kolmogorov: 1950, Foundations of the Theory of Probability (Chelsea Publ.'Co., New York, 1950). We take a field of propositions to be a set of propositions closed under negation and disjunction.

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RALPH KENNEDY AND CHARLES CHIHARA

Because o f the way some subjectivists specify the basic probability axioms, it may seem that the conditions should not be stated in terms o f logical necessity (i) and logical incompatibility (ii). For example, in his book (op. cit.), de Finetti defines the symbol 't--E' to mean 'E is certain' (p. 39) and then stipulates that 'events A and B are incompatible' is to mean 'I---~(A & B)', i.e. it is certain that not both A and B (p. 41). Thse definitions suggest that, for de Finetti at least, incompatibility o f events is mere 'subjective incompatibility'. However, such an interpretation o f the axioms of probability (and hence of 'admissibility') would lead to the abandonment of the Dutch Book Theorem. It is a simple exercise to show that admissibility, in this weakened sense, is neither a necessary nor a sufficient condition for coherence. Another way o f defining 'probability measure', which permits logically necessary propositions to have measure less than one, has been proposed by John Vickers in his 'Some remarks on coherence and subjective probability', Philosophy of Science (1965), pp. 3 2 - 3 8 . Vickers' version of condition (i) is stated: (i')

l f p is in ~ " and there is a proof of p, then f(p) = 1.

Here, 'there is a proof o f p ' should not be understood as meaning: p is provable. This can be seen from the following. Vickers justifies his version of condition (i) by claiming (without any justification) that the concept of coherence should be a decidable concept. He then suggests that since the concept o f proof is decidable, condition (i') does not preclude the decidability of coherence as does (i). Since it is wellknown that provability (even in first order logic) is undecidable, we conclude that Vickers probably meant by 'there is a proof of p ' something like: someone, somewhere, actually has a proof o f p. (Note: there are specific passages in the article that support this conclusion.) It should be mentioned that Vickers' reasoning above is questionable. Even if there is a decision procedure for determining if something is a proof, it is not obvious that there is a decision procedure for determining if someone, somewhere, has a proof o f p . In any case, the reader can verify that changing the definition o f 'probability measure' (and hence of 'admissibility') in either o f the above ways will not significantly affect our arguments against the DBA. s Our use of the term 'admissible' is consistent with de Finetti's use o f 'admissible assignment' in 'Foresight', p. 104, and is loosely based on Wesley Salmon's use o f the term in his: Foundations o f Scientific Inference (University o f Pittsburgh Press, Pittsburgh, 1967), pp. 53/4. This is not the criterion Jackson and Pargetter explicitly suggest. They take DB's to be accurately measured by FBQ's (pp. 403/4). But this simply will not, in general, do. If the betting situation is n o t competitive - if, say, clause (iii) o f our definition of a CBS does not hold - then there just w o n ' t be any good reason to suppose that X ' s FBQ's will match what we intuitively would take his DB's to be. This seems to be the most widely accepted use o f the term 'coherent'. But there are other uses. Abner Shimony, in his 'Coherence and the axioms o f confirmation', Journal of Symbolic Logic (1955), pp. 1 - 2 8 , calls a quotient incoherent only if there are choices Y could make such that "...X can at best lose nothing, and in at least one possible eventuality he will suffer a positive loss" (p. 9). And D. H. Mellor, in his: The Matter o f Chance (Cambridge University Press, Cambridge, 1971), uses 'coherent' idiosyncratically to mean what we mean by 'admissible' (p. 40). , The version o f the DBT cast in terms o f Shimony's notion o f coherence needs a correspondingly stronger sense o f 'admissibility', which Shimony provides (op. cit., pp. 5/6). His proof supports a version o f the DBT stated in terms o f conditional betting quotients. For simplicity, we work (as do Jackson and Pargetter) with a version involving unconditional betting quotients. For a proof of this version of the DBT see de Finetti's 'Foresight', pp. 103/4. 9 There is an anticipation o f the DBA in F. P. Ramsey's classic paper 'Truth and prob-

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ability', reprinted in Kyburg and Smokler, where he writes (p. 80): "If a n y o n e ' s mental condition violated these laws [of probability], his choice would depend on the precise form in which the options were offered him, which would be absurd. He could have a book made against him by a cunning bettor and would then stand to lose in any event." 10 See Ramsey, p. 74. 11 I ~ u s , de Finetti writes: "However, there are also reasons for preferring the opposite approach; the one which we a t t e m p t here. This consists in setting aside, until it is expressly required, the notion of utility, in order to develop in a more manageable way the study of probability", Theory of Probability, Vol. I, p. 80. 12 The betting behavior approach makes the simplifying assumption o f "the identity of m o n e t a r y value and utility" (de Finetti, ibid., p. 82). Such an assumption - that the utility o f m o n e y is linear - requires a limit to the size of bets that can be made (ibid., p. 93). 13 We owe this example to Roger Rosenkrantz. 14 The above example is based on an idea supplied by Ernest Adams. is See, for example, Mellor, Chap. 2. 16 The F o u n d a t i o n s of Statistics, 2nd ed. (Dover, New York, 1972), p. 28. 17 'Foresight', p. 102. 1~ ( . ) , taken literally, is highly questionable, since X might be in a CBS and yet n o t set FBQ's that m a t c h his DB's because, for example, he makes a slip o f the pen (or tongue) in setting his FBQ's or because he confuses one proposition to be bet on with another that is similar b u t n o t equivalent. Since (a-2) only requires that X be 'disposed' to choose matching FBQ's, such objections cannot be raised against it. 19 F o r t h c o m i n g in 'Application o f the Jeffrey decision model'. s0 This ambiguity in the UP was noticed independently by Ellery Eells and the Wake Forest author. ~1 For example, there is the a r g u m e n t from intuitively plausible principles of decision theory set forth by Savage in his: F o u n d a t i o n s of Statistics. There are also the arguments from axioms expressing ideal ordering relations of degrees o f belief. See, in this regard, * de Finetti, 'Foresight', p. 100/1 and Bernard K o o p m a n , 'The bases o f probability', reprinted in Kyburg and Smokier, op. cir.