2.4 Van Fraassen’s arguments against IBE
2.4.2 The inconsistency objection
Van Fraassen (1984, 1989) sets out to demonstrate that the practice of giving a bonus to explanatory hypotheses leads one to accept bets which one is guar-anteed to lose. His strategy is based on constructing a Dutch Book strategy – that is, a diachronic Dutch Book – against an agent who, in addition to the classical probability calculus, also adopts IBE as a procedure to update his beliefs.10
Let us explain the key terminology. ADutch Book is a set of bets offered to an agent by a clever (though mischievous) bookie, bets which have the following characteristics: (i) the bookie who generates them only knows the agent’s degrees of belief; (ii) each of the bets is accepted as fair by the agent;
and (iii) the set of bets guarantees that the bookie will garner a net win.
If an agent is liable to Dutch booking, then his degrees of belief violate the probability axioms.11 For that reason, a necessary condition for rational belief is taken to consist in an immunity to Dutch Books.
Van Fraassen (1989) distinguishes between two varieties of Dutch Books:
synchronicanddiachronic. Synchronic Dutch Books are known as Dutch Book Arguments and consist of sets of simultaneously offered bets, at one moment in time. By contrast, diachronic Dutch Books – known as Dutch Book strategies – rely on the bookie’s option to offer new bets at later moments. As already mentioned, to construct a Dutch Books Argument is equivalent to showing that an agent’s degrees of belief violate the axioms of probability calculus; the agent holds inconsistent beliefs, which is irrational. Yet, it will be shown that the same cannot be said about Dutch strategies: immunity from them is not required for rational belief.
Van Fraassen criticizes IBE for making those who follow it liable to Dutch Book strategies. He (van Fraassen 1989: 166–9) proceeds by imagining a series of bets between Peter and his Bayesian friend, Paul, with respect to a die whose bias of coming upace is not known. Assuming that the first four tosses of the die come upace(propositionE), the bookie (Paul) proposes the following bets concerning the hypothesis H that the fifth toss will be also ace:
Bet I pays $10,000 if E is true and H is false.
Bet II pays $1,300 if E is false.
10More recently (1995), van Fraassen has ceased to rely on Dutch Books as a means to discuss rationality issues.
11As John Earman (1992: 39–40) points out, the Dutch Book theorem proves that if any of the probability axioms is violated, then a Dutch book can be made. TheConverse Dutch Book theorem proves that if the axioms are satisfied, then a Dutch book cannot be made in a finite number of steps. Earman also indicates the difficulties with the Dutch Book justification of the probability axioms.
Bet III pays $300 if E is true.
The bookie bases his proposal on his knowledge of the fact that Peter has learned IBE from a travelling Preacher (reason for which van Fraassen calls IBE the “Preacher’s Rule”; it says that the posterior probability for each hypothesis of bias depends not only on the initial probabilities and the series of outcomes, but also on their explanatory success). Thus, Paul knows that Peter will assign a higher value to the probability that the fifth toss will also come upace than he would if he only followed Bayesian conditionalization.
Peter assesses the fair costs of the bets on the basis of the outcomes’ prob-abilities and the offered values. The model used to calculate initial prob-abilities introduces a factor X of bias, which can take N different forms:
X(1), . . . , X(N). If the die has bias X(I), then the probability of ace on any one toss isI/N. X(N) is the perfect bias, givingace a probability (N/N) = 1 (cf. van Fraassen 1989: 163). In Peter’s model,N = 10:
P(E) equals the average of (.1)4, . . . ,(.9)4,1, that is.25333.
P(¬E) is.74667.
P(E&¬H) is the average of (.1)4(.9), ...(.9)4(.1),0, that is.032505.
These probabilities, along with the values of the bets, give the following costs of the bets, which both Peter and the bookie consider to be fair:
The fair cost of bet I is $325.05.
The fair cost of bet II is $970.67.
The fair cost of bet III is $76.00.
The total cost of the bets is $1,371.72. Since he deems them fair, Peter buys them all from his friend. Now, van Fraassen continues, suppose that not all four tosses have come up ace, that is,E is false. So Peter loses bets I and III and wins bet II. In other words, Peter spends a total of $1,371.72 and receives
$1,300. Hence the bookie makes a net gain of $71.72.
IfE has come out true, Peter loses bet II, but has already won bet III, so he gets $300 from the bookie. At this point, bet number I can be formulated as
Bet IV pays $10,000 if H is false.
The bookie proposes to buy this bet from Peter, who agrees to sell it for
$1000, because his probability that the next toss will be anace is, as dictated by Preacher’s Rule,.9. The moment Peter pays $1000 for bet IV, Paul can be happy: even before the fifth toss, he has a guaranteed gain. He paid $300 for
losing bet III and $1000 for buying bet IV, i.e. $1300, which ensures him a net gain of $71.72.
So, van Fraassen concludes, as a belief-updating rule, IBE leads to inco-herence:
What is really neat about this game is that even Peter could have figured out beforehand what would happen, if he was going to act on his new probabilities. He would have foreseen that by trading bets at fair value, by his own lights, he would be sure to lose $71.72 to his friend, come what may. Thus, by adopting the preacher’s rule, Peter has becomeincoherent – for even by his own lights, he is sabotaging himself. (van Fraassen 1989: 168–9)
Let us now analyze van Fraassen’s argument against IBE. Kvanvig (1994) reconstructs it neatly:
(1) There exists a series of bets, described as bets I–IV above, with a cost as noted above, and that, if taken, guarantee a net loss no matter what happens.
(2) If one is a consistent follower of a particular ampliative rule, one regards this set of bets as fair.
(3) It is irrational to regard such a series of bets as fair.
(4) Therefore, it is irrational to be a consistent follower of the particular ampliative rule that implies than one regards all of the bets in question as fair.
(5) For any ampliative rule, there is a set of bets that are regarded as fair if one consistently follows that ampliative rule which constitutes a Dutch Book strategy.
(6) Therefore, it is irrational to be a consistent follower of any ampliative rule. (Kvanvig 1994: 331)
The weak point of this argument is premise (3). In fact, it is doubtful whether it is true. Moreover, as will be seen, contrary to van Fraassen’s assumption, the bookie actually did cheat against Peter.
Note that not all sets of bets which guarantee a net loss are irrational. To see this, let us consider two categories of bookies with extraordinary powers, whom, inspired by Christensen, we shall term Super-bookies: (i) the Omni-scient Super-bookie, and (ii) the PreOmni-scient Super-bookie.
(i) The Omniscient Super-bookie knows everything. In particular, he knows with certitude the truth-values of every proposition which can make the object of a bet. When buying a bet from him, the only chance of a human fallible agent is to assign a probability of either 0 or 1 to any proposition, and to be correct about it. However, for fallible agents, such a doxastic practice would be irrational. Rationality forbids the ascription in all situations of extreme probability values to the given propositions. Thus, our agent is sure to incur a loss.
(ii) The Prescient Super-bookie is the one who has privileged information about the agent’s probability distribution. More exactly, he knows not only the current probability distribution, but also what changes the agent will make in his degrees of belief over time. Here is what could happen if this was the case:
...suppose the prescient bookie knows that one’s probability forptoday is .7, and also knows that tomorrow it will be .6. In such a case, a series of bets guaranteed to net one a loss is easy to construct: the bookie offers, and you accept, a bet today that pays $10 and costs .7($10) = $7, if p is true, and buys a bet from you that pays $10 if p is true, and costs .6($10) = $6. Then you and your bookie exchange a $10 bill whether or notpis true, resulting in a net profit for the bookie of $1. If this bookie knows your probability will be higher tomorrow than today, he employs the same strategy, this time offering and buying bets on the negation of p. (Kvanvig 1994: 332–3)
Therefore, the agent again is guaranteed a net loss if he buys bets from the Prescient Super-bookie.
The possibility to block such a Dutch Book strategy seems to consist in obeying a strong diachronic condition which precludes any doxastic modifi-cation whatsoever. Christensen labels it the “Calcifimodifi-cation” condition. No doubt, Calcification cannot be a reasonable requirement. There are numerous cases in which rationality urges us to modify our credences. For that reason, as Christensen puts it, “we do not think that the beliefs an agent holds at dif-ferent times should cohere in the way an agent’s simultaneous beliefs should.”
(Christensen 1991: 241). A Dutch strategy would indicate the inconsistency of someone’s set of beliefs only if those beliefs ought to be consistent. Often, in light of new relevant information, we do over time arrive rationally at beliefs contradictory to (not to mentiondifferent from) the earlier ones.
It actually seems that the rational thing to do is simply not to bet against Super-bookies. If an agent did so in spite of information about the extraor-dinary powers of the bookie, then we say that he acted stupidly. Rationality urges us to be suspicious about such bookies. The point is also expressed by Christensen:
There is, after all, no Evil Super-bookie constantly monitoring everyone’s credences, with an eye to making Dutch Book against anyone who falls short of probabilistic perfection. Even if there were, many people would decline to be at “fair odds”, due to suspiciousness, or risk aversion, or re-ligious scruples. In short, it is pretty clear that Dutch Book vulnerability is not,per se a practical liability at all! (Christensen 1991: 237)
Kvanvig makes the important observation that Paul is in fact a kind of pre-scient bookie. What Paul knows is the ampliative rule which Peter follows.
This also requires having some crucial knowledge about the future, knowledge which has produced the set of bets that incurred Peter a net loss. Had Peter followed a different ampliative rule from the one he actually did, then the set of bets offered by the bookie would have failed to be a Dutch Strategy. For instance, as Kvanvig calculates, if, after finding out thatE obtains, Peter had raised his probability ofH not to .9, but to .88. then bet IV would have failed to assure the bookie a net win. Of course, as Kvanvig (1994: 347) points out, in that case a different set of bets could have been offered so as to ensure a net win for the bookie. But in order to generate this different set of bets, the bookie would have needed to know Peter’s different ampliative rule.
The conclusion is that since the Dutch strategy conceived by Peter’s friend qualifies him as a Prescient Super-bookie, we may conclude that (3) is false:
there is nothing about van Fraassen’s Dutch Book to imply that Peter was irrational. What van Fraassen needs is a Dutch Book strategy which does not rely on privileged knowledge about the future decisions of the follower of the Preacher’s rule.
Naturally, we should do justice to van Fraassen’s remark that Peter is sabotaging himself evenby his own lights. Yet the remark is not specific about what is necessitated in order to equate being subject to a Dutch strategy with irrationality. I believe it is transparent that the additional requirement on rationality that he has had in mind is his principle ofReflection.12 Here is how he has formulated it:
To satisfy the principle, the agent’s present subjective probability for propositionA, on the supposition that his subjective probability for this proposition will equalrat some later time, must equal this same number.
(van Fraassen 1984: 244)
12Here is where I believe that van Fraassen turned to the Reflectivity principle: in dis-cussing the notion of rationality in a practical context, he states that “a minimal criterion of reasonabless if thatyou should not sabotage your possibilities of vindication beforehand.”
(van Fraassen 1989: 157). It emerges from the context that he has in mind an epistemic kind of vindication.
The principle be can formalized as follows:
Ref lection:P0(A|P1(A) =r) =r,
where P0 is the agent’s present probability function, and P1 her probability function at some future time. In intelligible words, the principle demands that
“if I am asked how likely it is to rain tomorrow afternoon on the supposition that tomorrow morning I’ll think rain 50% likely, my answer should be “50%”.
(Christensen 1991: 232).
Nonetheless, Christensen constructs several cases to show intuitively that rationality actually urges irreflectivity (i.e. violation of the Reflection principle) in van Fraassen’s sense. He imagines, for example, that a psychedelic drug, which he calls LSQ, has the power to induce those under its influence the strong conviction that they can fly. It can be also assumed that the drug does not diminish in any sense the capacity to reason and to understand. If the agent believes that he has just taken a dose, and someone asks him, “What do you think the probability is that you’ll be able to fly in one hour, given that you’ll then take the probability that you can fly to be .99?”, he must, following the Reflectivity principle, answer “.99”. Obviously, this is absurd. Our agent knows about both the properties of LSQ, and the fact that the drug cannot confer the power to fly. Therefore, rationality impels him to violate Reflection.
Other common-sense examples of rationally requiring irreflectivity involve propositions such that my coming to have a high degree of belief in them would in itself tellagainst their truth. An example is the proposition that I have no degrees of belief greater than .90. Christensen then asks,
What credence should I accord this proposition, on the supposition that I come to believe it to degree .95? Reflection says “.95”; elementary probability theory says “0”. This seems to be a case where Reflection cannot be a “new requirement of rationality, in addition to the usual laws of probability” – it is inconsistent with those very laws. (Christensen 1991: 237)
On balance, I do not see so far how vulnerability to Dutch Book strategies generally implies irrationality.
Finally, I follow Kvanvig in raising one last counterargument against van Fraassen’s Dutch Book strategy. We saw, on the one hand, that when presented with bet IV, Peter assigned, in virtue of Preacher’s rule, a probability of .9 for the fifth toss to come outace. However, Peter applied this rule only after having learned thatE is the case. That is to say, he calculatedP(E),P(¬E), and P(¬H & E) by usual probability calculus. But he should have done the same with respect to the fifth toss. He should have calculated P(H |
E) = P(H & E)/P(E), where P(H & E) was to be obtained as the average of (.1)5, . . . ,(.9)5 (which is equal to .22). Given the already known value of P(E) =.25333, Peter would have obtained forP(H |E) the value .87. Instead, he applied the Preacher’s rule going in to the fifth toss, which induced him to believe that the probability of getting a fifth ace is .9. The moral is properly formulated by Kvanvig:
Van Fraassen’s argument therefore trades on having Peter following IBE advice after learning E, but ignoring that advice prior to learning E.
That, however, is unfair to IBE; if IBE is problematic, what must be shown is that a consistent application of an IBE strategy is subject to Dutch Book difficulties. (Kvanvig 1994: 338)
Peter lost because he committed himself to values inconsistent with those dictated by the Preacher’s rule. He should not have accepted bets whose costs had been generated by calculations other than the ampliative procedure that he usually followed.
We can now summarize the counterarguments to van Fraassen’s Dutch Book strategy against IBE. We saw first that there are clear cases of Dutch strategies where no irrationality is involved, namely when the agent bets against Super-bookies. We further saw that van Fraassen’s Reflection prin-ciple, which he conceives as a consistency constraint on the agent’s diachronic probability distribution, can and should in some situations be violated as a matter of rational behavior (e.g., the LSQ case and the case of “self-blocking”
propositions). Finally, we have indicated that van Fraassen’s argument against IBE is not exactly fair. This should be enough to warrant the conclusion that van Fraassen’s Dutch strategy against IBE has missed its target.