Van Fraassen’s Dutch Book Argument against Explanationism
Dorit Ganson
Quiet, don’t explain
What is there to gain
Skip that lipstick
Don’t Explain
Billy Holiday
-
As I am reminded by my 4 year old daughter, who loves to ask strings of why questions to the point where
our explanatory resources give way, the desire for explanation, the need to make sense of the world in order to
understand it and change it, is deeply human and, for most people (save a few constructive empiricists such as Bas
van Fraassen) irresistible. Sadly, our desires often exceed our abilities to satisfy them, and we wind up fabricating
explanatory stories far beyond the limits of reason alone. And even when we are faced with explanations that seem
genuinely satisfying to us, it is far from obvious that there is something good and general to say about what
epistemic attitude we ought to take to them. Against the backdrop of a growing skepticism about the notion that
philosophers have special resources for understanding what the aims of human inquiry really are or should be, as
well as what we need to do or think, or not do or not think, in order to help us get there, or never get there, the
prospects for acquiring specifically philosophical insight into whether or when to believe our best explanations do
seem grim.
Despite worries about the legislative role of epistemology, commitments concerning the epistemic status of
explanatoriness—its relevance to a theory’s likelihood of truth and rational acceptability—are quite widespread in
contemporary philosophy, often under the guise of an expressed attitude towards “inference to the best explanation.”
Though there is little by way of positive defense of the legitimacy of such a form of inference among those who
believe in it, various substantive projects in epistemology and metaphysics rely on this variety of reasoning. 1
Nowhere is controversy over the epistemic force of explanatoriness quite as lively and significant as in the
debates over scientific realism, where whether you accept or reject IBE (under various interpretations of what such
stances really amount to) seems to determine which side of the conflict you fall on. Despite the plurality of
approaches and aims in work done on behalf of scientific realism, one unifying theme is that rationality sometimes
demands that we regard the best explanation of puzzling phenomena as more likely to be true than explanatorily
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inferior rivals—even those which are equally empirically adequate.2 Anti-realism, on the other hand, involves the
view that, in so far as explanatoriness trades on features of theories that go beyond considerations of logical
consistency and empirical adequacy, we ought to regard explanatoriness as epistemically irrelevant: it is never a
good reason to increase our confidence in a theory’s truth. Such a virtue in a theory speaks only to its pragmatic
advantages. So, for a realist, there will be episodes in scientific inquiry where the explanatory strength of a
particular theory or hypothesis should affect our conception of its likelihood of truth; for an anti-realist, it should
not.
Since a fair amount of substantive philosophical work depends on accepting explanatoriness as a
potentially epistemic virtue (a view I shall call “explanationism”), it would be nice to have a clearer sense of
whether this view is at all defensible. Can we even begin to answer the most weighty and seemingly decisive
arguments against it?
We shall focus here on the most devastating of the bunch, van Fraassen’s clever Bayesian Peter argument,
which appears in his second major anti-realist tract, Laws and Symmetry. Recounting the travails of hapless Peter,
who is so foolish as to heed the advice of an evangelist for inference to the best explanation, van Fraassen tries to
show that anyone who allows explanatoriness to enhance her degree of credence is susceptible to a Dutch book. 3
This sorry predicament indicates that accepting explanationism leads to an unacceptable incoherence in one’s belief
state.
In van Fraassen’s parable, Peter has fallen under the influence of an explanationist preacher who urges him,
as a general rule, to increase his epistemic confidence in hypotheses which emerge as best explanations. Peter is
confronted with an alien die with a bias of any one of X(1), X(2)...X(10), he knows not which (a bias of X(n)
indicates that the probability of ace on any single proper toss is n/10). Given outcomes E: the die coming up aces
for four consecutive tosses, and H: the die comes up ace on the fifth toss, his opinions about the likelihood of E, and
of E and -H are determined by taking an equally weighted average of the probability of these outcomes for each of
the ten possible scenarios, bias X(1), bias X(2), etc.. A friend comes along, and they negotiate bets on E & -H, -E,
and E which Peter regards as fair since their price is properly based on his probability assignments. In the course of
the game, a string of four aces is tossed. Thinking that the best explanation of this occurrence is a hypothesis of
high bias, and mindful of the preacher’s urgings, Peter raises his current probability for the event “the fifth toss
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shows ace” turning out true from .87, the result obtained from a routine application of Bayes’ Theorem, to .9. After
further betting negotiations with his friend, which again are fair according to Peter’s updated probability
assignments, he winds up with a Dutch book. 4
Thanks to the influence of the IBE preacher, Peter winds up
with a Dutch book. Now that he has sold bet I, he must count his net losses at 1,071.72 - 1,000 = 71.72. He could
have seen at the beginning of the game that he was bound to lose 71.72, whether or not E and H turned out to be true
or false. Van Fraassen triumphantly concludes: “What is the moral of this story? Certainly, in part that we
shouldn’t listen to anyone who preaches a probabilistic version of Inference to the Best Explanation, whatever the
details. Any such rule, once adopted as a rule, makes us incoherent.” (van Fraassen 1989, p. 169)
There are a number of possible responses to van Fraassen’s argument, several of which are suggested, if not
always fully developed, in the fairly extensive literature which aims to confront his criticisms of inference to the best
explanation and scientific realism. One tends to find in this literature an underestimation of the multiplicity of
defensive options, as well as a failure to acknowledge the deeper problems inherent in the project of trying reconcile
explanationism with Bayesian accounts of confirmation—problems which remains even after the shortcomings of
van Fraassen’s way of demonstrating the diachronic incoherence of explanationism have been exposed. First,
however, we shall examine the wide array of potential solutions to the Bayesian Peter objection.
1. Peter’s Carelessness
Douven (1999), Kvanig (1994) and Ganson (2001) all explore some version of the criticism that Peter
doesn’t seem to look before he leaps: it is far from clear that a more conscientious Peter–one who is more alert to
what his rules for belief revision actually are–would accept his friend’s initial bets as fair. Van Fraassen
nonchalantly states at the beginning of the tale that, before any evidence comes in, Peter will calculate his personal
probability for E&-H, the object of Bet I, by taking the average of p(E&-H/bias X(1)),...,p(E&-H/bias X(10)). This
result follows from a simple application of the theorem of total probability since the mutually exclusive and jointly
exhaustive hypotheses of bias are initially equiprobable, each having probability 1/10:
Using pn(Y) as shorthand for p(Y/bias X(n)),
p(E&-H) = p1(E&-H)p(bias (X1)) + ... + p10(E&-H)p(bias( X10))
= (i/10)4(1-i/10) = .032505
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Since p(-H/E) = p(E&-H)/p(E) = .032505/.25333 = .13, Peter has already essentially committed himself to
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a value for this conditional probability–a value at odds with the assignment he offers later on in the story, p(-H/E) =
.1. For Peter, once E happens, the hypothesis of high bias gets “bonus points” on account of its explanatory success,
and this increase in credence in turn affects his personal probability for the fifth toss being, or not being, an ace.
Van Fraassen’s calculation of p(E&-H) on Peter’s behalf, one could suggest, takes no accounting of the
fact that Peter is the sort of explanationist whose probability values for fifth toss events like H or -H will be affected
by whether or not E happens. As such, the calculation is not really true to the rules which govern Peter’s degree of
belief assignments. Given the necessary interdependence between p(E&-H) and p(-H/E), any determination of
p(E&-H) will somehow have to incorporate Peter’s explanationism as well, a requirement which is violated by van
Fraassen’s method, the averaging of values for pn(E)pn(-H/E) given equally weighed biases.
Who knows exactly what peculiar methods or rules Peter would want to use to set a value for p(E&-H), but
we do know from what happens towards the end of the tale that p(-H/E) = .1. Consequently, supposing Peter would
still set p(E) = .25333, we could picture him assigning
p(E&-H) = p(E)p(-H/E) = (.25333)(.1) = .025333. Peter’s fair price for Bet I, which pays $10,000 if E&-H, would
then be $253.30. If we use this value, and follow along with the rest of Peter’s adventures, he would end up gaining
3 cents.5 Three cents is a fairly insubstantial amount, but it is enough to save him from the perils of his friend’s
Dutch book.
In the discussion before the Bayesian Peter story is presented, van Fraassen creates the misleading
impression that his argument will show that any “ampliative” rule of inference to the best explanation leads to
incoherence: allowing explanatoriness, and not just predictive capacity, to factor in as a feature relevant to revising
our degree of belief as evidence comes along will, in general, be irrational. The Peter of the original parable is
hardly an adequate representative of this sort of broad explanationist view. Instead, according to the response we
have been exploring, he turns out to be a particularly unlikely sort of philosophical victim whose real problem is not
his belief in the relevance of explanatoriness to likelihood of truth, but rather his tendency not to think very carefully
ahead of time of what his rules for degree of belief revision actually are. If the Peter in van Fraassen’s version of the
story had been attentive to his own rules for updating his probability function in light of evidence, his
explanationism wouldn’t have been ignored in his determination of p(E&-H), and he would never have been duped
in the first place.
4
Van Fraassen almost seems to anticipate this kind of response in a brief discussion of the preacher’s
doctrine which precedes the main argument. Here, the preacher protests “Your problem was that you did not give
up enough of your pagan ways. This probability of .87 had already become quite irrelevant to any question of what
would happen after an initial run of four aces. If you had followed the rule properly, you would not only have
foreseen that your new probability afterward would become 0.9. You would also now have had as conditional
probabilities, those probabilities that you would get to have, after seeing such an initial run of aces. Thus the
leavening effect of explanatory power in our lives, although activated properly only after seeing explanatory
success, reaches back from the future into the present as it were, and revises our present conditional opinions.” (van
Fraassen 1989, p. 167).
Peter’s friend apparently dodges these protestations by, in van Fraaassen’s words, being “careful not to
propose any conditional bets” as he generates a Dutch book against Peter. But these maneuvers only superficially
avoid such bets, since the combination of bets I and II is equivalent to a conditional bet on -H given E which costs
$1300. Van Fraassen himself admits as much in a different context when he tries to show that violating the principle
of reflection leads to incoherence. Of a bet I which pays y if -H & E and a bet II which pays yx if -E, where
x = p(-H & E)/p(-E), as in the Bayesian Peter example 6x = p(-H & E)/p(E) = .13, and the cost of bet II = xy = 1300.
(the only difference being that in the reflection argument y=1, so the cost of bet II is simply x), he says: “It helps to
observe that I and II together form in effect a conditional bet on -H on the supposition that E, which bears the cost x
and has prize 1, with the guarantee of your money back should the supposition turn out to be false.”(van Fraassen
1984, p. 240).7yxP(E) + xyP(-E) = xy (P(E) + P(-E)) = xy. If -E happens, the agent wins her money back.
2. An Implausible Explanationist Rule
We should hardly expect that a real explanationist would, or would have to be, as foolhardy as Peter.
Ignoring your own rules for probability function revision at the beginning of betting even though they are relevant to
assessing subjective probability values for particular events which form the object of bets, and then letting these
rules influence your probability assignments and choice of further betting arrangements later on (in a predictable
way) clearly creates opportunities for clever bookies. What’s more, even if Peter had been more attentive to his
updating rules from the very start in the manner suggested above, the results are far from satisfying. Peter’s
explanationist principle seems to commit him to a somewhat perverse value for P(-H & E), one wholly at odds with
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a straightforward, well justified method for determining prior probability assignments. There’s nothing inherently
incoherent about setting priors in an unreasonable or wacky way, though of course assignments which satisfy the
minimal formal requirements of probabilistic coherence can still be criticized on other grounds. Could a sensible
explanationist ever be drawn to Peter’s rule?
The rule in question is highly general: the explanatoriness of a hypothesis should make you increase your
degree of credence, even beyond what would be recommended by what could be called a “neutral” application of the
probability calculus. When you update your probability function in an alien die case, for instance, where the
competing hypotheses high bias, low bias start out with the same prior probability, you should not simply take into
account how each hypothesis weighs the evidence, i.e. how probable the evidence is under the assumption that the
hypothesis is true; you should add on bonus points once a hypothesis becomes explanatory. This increase in
credence will in turn lead to additional increases or decreases in your probability assignments for events (such as, in
the case at hand, the fifth toss shows ace) which are statistically affected by whether or not the hypothesis is true.
The credence boost in this instance seems highly unmotivated. When we actually make a judgment about
which hypothesis, high bias or low bias, is the best explanation in light of the evidence, there is no information
relevant to our decision beyond how each hypothesis weighs the evidence. All that grounds the explanatory
superiority of the high bias is its greater predictive success, so we would hardly expect to be able to raise our
credence over and above what is suggested by a routine, neutral application of Bayes’ theorem.
The most reasonable explanationists will recognize that there are many cases, especially in normal science,
where hypotheses—all with the same prior probabilities—are essentially distinguishable by the different weights
which they accord to various sorts of outcomes. In such cases, the best explanation will simply be the hypothesis
which achieves the greatest predictive success. Special bonus points for being the best explanation will not be added
(though the best explanation will still be taken to be more likely to be true than the alternatives on normal Bayesian
grounds). The reasonable explanationist, then, would not agree that we should follow Peter’s peculiar rule.
This is not to say that the explanationist would never recommend a boost in credence beyond what would
be suggested if we simply took predictive success into account. Explanatory superiority is a property sometimes
wholly grounded in greater predictive success, sometimes not. When it is not (and there are different views about
what more it could consist of) we might find ourselves with additional information which is relevant to our
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probability assignments. That the explanationist would feel that such a situation can arise is particularly clear when
we look at the sorts of examples which most notoriously divide realists and anti-realists: cases where we are faced
with incompatible, but equally empirically adequate theories with different degrees of explanatoriness. The
realist/explanationist, against the anti-realist/anti-explanationist, will insist that there are cases where we are
sometimes rationally constrained to regard the best explanation as more likely to be true than the available
alternatives, even when these rivals are equally empirically successful.
A somewhat similar response to the Bayesian Peter objection is suggested in Ben-Menahem (1990). BenMenahem, like Lipton (1991), Day and Kincaid (1994) and Douven (1999), stresses the importance of
acknowledging the contextual character of explanatoriness. In some contexts, explanatory superiority is judged on
the basis of what he calls “structural” considerations (by which he appears to mean highly general, formal features
of theories, such as simplicity); in others, notably non-philosophical ones, judgements of explanatory superiority are
made on the basis of appeal of to “broad empirical considerations” (general facts or claims which have been
sanctioned by empirical research.): the best explanation is the hypothesis judged to be most credible in light of our
background empirical knowledge. Only in the latter contexts is explanatoriness linked to likelihood of truth, and so
only the in these contexts does inference to the best explanation genuinely and legitimately enter into our
deliberations.
The problem with van Fraassen’s use of the Bayesian Peter story to illustrate the flaws of inference to the
best explanation, according to Ben-Menahem, is that the context in question is one where broad empirical
considerations play no role in the determination of the best explanation: which hypothesis is the best explanation in
the end is simply a matter of which hypothesis accords the highest probability to the data. Purely formal criteria are
involved in determining the best explanation, so adding bonus points beyond what purely formal probabilistic
considerations dictate is unwarranted. Inference to the best explanation doesn’t really enter into the picture.
“...when the only criterion for evaluating the explanatory power of a hypothesis is the degree of probability
conferred by that hypothesis on the data, one cannot use this evaluation of explanatory power to redistribute the
probabilities...I welcome van Fraassen’s argument for it adds further weight to a point I have stressed several times
above: the rationality of IBE depends on the standards we use to assess explanatory power. Where there are not
explanatory merits except the structural ones, there is no room for the application of IBE.” (Ben-Menahem 1990).8
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Though Ben-Menahem is right to insist that the context of the Bayesian Peter argument is one where bonus
points are not warranted, his justification for this claim is somewhat problematic. Ben-Menahem suggests that
inference to the best explanation is not really involved when determining explanatory superiority is just a matter of
judging which hypothesis accords the highest probability to the data. This move seems ill-advised. As we shall see
in the next section, some explanationist philosophers, such as Harman, think that it is precisely in the assignment of
likelihoods (the conditional probabilities p(e/Hi)’s) and prior probabilities (p(Hi)’s) that considerations of
explanatoriness find their expression. The best explanation, in any context, will be the hypothesis which wins out
over others with respect to the dual consideration of its initially plausibility, and the weight it accords to the
evidence which actually happens (in accordance with Bayes’ theorem). Even if it should turn out that Harman’s
suggestion cannot accommodate all cases where explanatoriness bears on one’s degree of belief, we can grant that
many cases will nicely fit his proposal: e.g., after I cough up bloody sputum, the best explanation for my continuing
cold symptoms is that I have a secondary bacterial infection, instead of simply a viral infection since bloody sputum
is more likely given the former hypothesis than given the latter hypothesis. I see no reason to deny that this case
involves inference to the best explanation, even though which explanation wins out is simply a matter of
determining which hypothesis accords the highest probability to the data at hand.
Ben-Menahem further distinguishes the Bayesian Peter case from contexts where inference to the best
explanation is genuinely and legitimately involved by noting that, in the Peter case, no background empirical
considerations or generalizations are invoked in deciding which hypothesis is most credible. He seems to suggest
that because only purely formal, a priori considerations play a role in the relevant probability assignments, inference
to the best explanation does not or should not really enter into Peter’s deliberations. But even this conclusion is
subject to some doubt. Peter does implicitly rely on substantial background empirical commitment: for example, he
presupposes that there are no environmental factors (magnetic fields, hidden wires and such) which might distort the
frequency of tossing an ace given a particular bias. This commitment has a bearing on his likelihood assignments,
and hence on his judgment of which hypothesis accords the highest probability to the data.
3. Put it in the Priors
Day and Kincaid (1994), Harman (1999), and Okasha (2000) also develop the criticism that van Fraassen
presupposes an idiosyncratic and uncharitable construal of inference to the best explanation. These authors all
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conclude that van Fraassen misidentifies the way attentiveness to explanatory factors could be reasonably thought to
enter into our deliberations. Rather than warranting a boost in credence after the evidence comes in, such factors,
first of all, may be crucial in our initial efforts to establish which candidate hypothesis should be taken seriously
(Okasha 2000), (Day and Kincaid 1999). The background empirical assumptions or specific causal commitments
which inform our judgments of explanatory strength help us to determine which hypotheses are worth considering,
as well as to decide when this set of alternatives is too limited, and what new options are available in light of the
limitations of previous theories. Second, we appeal to explanatory considerations to help us assign prior
probabilities and likelihoods. Though the authors who pursue this line of response have somewhat different views
on what these considerations involve (context-specific background knowledge of causal processes for Day and
Kincaid, constraints of conservatism, simplicity, and degree of data accommodation for Harman), all agree that
explanatory considerations help us determine the initial degree of plausibility of the available options, as well the
likelihood of certain data and experimental outcomes given the various alternatives. Any factor that would enter
into the evaluation of H as a better explanation than H' would be reflected in a higher prior probabilities and/or
likelihoods.
This effort to reconcile explanationism with Bayesian demands seems to work well enough for a wide
range of cases, but as we shall see after we look at one final response to van Frassen’s argument, another Bayesian
puzzle will expose its limitations.
4. The Illegitimacy of Diachronic Dutch Book Arguments
Even some hard core Bayesians, such as Howson and Urbach, believe that only synchronic Dutch Books—
Dutch books built up from simultaneous degree of belief ascriptions, and bets the agent simultaneously regards as
individually fair—highlight genuine failures of rationality. ‘Coherence’, according to this view, is a term which has
legitimate application just when used to describe belief states at particular points in time, as opposed to sets of belief
states over stretches of time. It is no more incoherent to make degree of belief assignments at different times which
correspond to bets which ultimately cannot be regarded as simultaneously fair than it is inconsistent to believe p at
one point, and -p later. The probability calculus, then, like the laws of logic, places rationality or coherence
constraints on the structure of our belief state at a given moment; both formal systems have much less to say on how
our belief state at one time should relate to our belief state at a later time to preserve rationality.
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On the assumption that this view is correct (an assumption I defend in Ganson 2001), we have yet another
reason for dismissing van Fraassen’s Bayesian Peter objection, which takes the form of a diachronic Dutch Book
argument.9
Leeds also criticizes van Fraassen for failing to explain why there is an epistemically relevant
difference between violating conditionalization by a rule, and not by a rule. But it should be clear why van Fraassen
thinks there is such a difference: in the former case, the bookie is in a position form the start to offer a series of bets
which lead to certain loss; in the latter, he is not. Unfortunately for the explanationist, however, we don’t really
need to rely on the charge of diachronic incoherence to show that trouble arises when explanationists try to respect
Bayesian principles.
5. Another Bayesian Puzzle
We don't need a complicated and somewhat misleading story about the follies of Bayesian Peter to see that
it might be difficult to conform to the probability calculus as evidence comes along once we allow explanatory
power to influence our judgments about likelihoods or likelihood comparisons. Examining Bayes’ Theorem:
p'(Hi) = p(Hi/e) = ___________p(e/Hi)p(Hi)____________________
p(e/H1)p(H1)+p(e/H2)p(H2)...+p(e/Hn)p(Hn)
we see that we can compare the p(Hi/e)’s just by looking at the p(e/Hi)p(Hi)’s, since the denominator is the same for
all the p(Hi/e)’s. Imagine a situation where:
i. H and H' are both accorded the same initial probability (say that prior to exposure to startling evidence, both are
considered equally likely)
ii. e is considered equally likely on H and H’: p(e/H) = p(e/H') (suppose e is a deductive consequence of both
hypotheses given certain background assumptions)
iii. we affirm that H is a better explanation for e than H'.
We might be tempted to say p(H/e) > p(H'/e) because of H’s capacity to explain e in a more satisfactory way, but a
quick glance back at Bayes’ theorem shows us that we cannot consistently do so.
The fact that (under the interpretation of the function we have been using) p(H/e) > p(H'/e) even though
p(e/H) = p(e/H') cannot always be grounded in differences in prior probability assignments to H and H', as Harman
et al. would suggest. Imagine that we regard e1 and e2 as incompatible, unusual, but possible events which are
equally likely. e1 and e2 are equally likely on rival theories H and H' which in turn are accorded the same
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probability, but H explains e1 better than H' explains e1 and H' explains e2 better than H explains e2 (perhaps H and
H' focus on different types of phenomena). We don't know what evidence we'll end up facing, so which hypothesis
is most explanatory can't be settled ahead of time and reflected in prior probability assignments p(H) and p(H').
Here we haven't a chance of tracing preferences based on explanatory power to prior probability assignments since
p(H) = p(H'). In this example, p(H/e1) > p(H'/e1), but p(e1/H) = p(e1/H') and p(H) = p(H').
We have noted that the realist/explanationist accepts that there are cases where greater explanatoriness
warrants a boost in credence above and beyond what we would assign if we took only predictive success or
empirical adequacy into account: we are sometimes rationally constrained to regard the best explanation as more
likely to be true than equally empirically adequate rivals. We are rationally constrained, for example, to think that
the best explanation for the phenomenon of Brownian motion (e1)—namely, the molecular hypothesis (H1)—is
more likely to be true than the hypothesis that matter is continuous (H2), even if this radical hypothesis could in
principle be incorporated into a non-discreet theory of matter which is empirically equivalent to molecular theory.
In other words, once we witness and accept e1 (p'(e1)=1), our revised probability function p' must assign a higher
value to H1 than H2:
p'(H1) = p(H1/e1) > p'(H2) = p(H2/e1), even though p(e1/H1) = p(e1/H2). If we accept Bayes’ Theorem, how is
this possible?
We have already explored one solution which will usually work quite well: accommodate for the
differences in value between p(H1/e1) and p(H2/e1), despite the equality of p(e1/H1) and p(e1/H2), by making the
p(H1) higher than p(H2). For example, the prior probabilities p(H1) and p(H2) may reflect H1 and H2’s differing
degrees of plausibility in light of constraints of conservatism, simplicity, etc., as Harman would suggest, or differing
degrees of fit with background empirical commitments and causal principles as Okasha, Day and Kincaid would
propose (again, independent of considerations of fit with data).10
We might come across a situation, however, where which hypothesis is ultimately most plausible in light of
the relevant background constraints and commitments will depend on what sort of empirical data comes along. Say
that two hypothesis H1 and H2 are, before the evidence comes in, equally plausible in light of background theory, or
equally recommended by our causal principles: p(H1) = p(H2). Two possible, incompatible outcomes of a particular
experiment are e1 and e2, and p(e1/H1) = p(e1/H2), p(e2/H1) = p(e2/H2), but H1 will fit better with background
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theory/causal principles than H2 if e1 happens, and H2 will fit better with background theory/causal principles than
H1 if e2 happens (H1 will emerge as the best explanation if e1 happens; H2 will count as the best explanation if e2
happens). Since greater explanatoriness should have a positive impact on our degree of credence, we want to say
that p'(H1) > p'(H2) if e1 happens, or p'(H2) > p'(H1) if e2 happens, but how can we square this probability
assignment with Bayes’ Theorem, given that all the relevant prior probabilities and likelihoods are the same? Must
we abandon our explanationist inclinations in this example and concede that p'(H1) = p'(H2)?
For the explanationist, whether H1 or H2 turns out to be explanatorily is as relevant to our degree of belief
revisions as whether e1 or e2 happens. It would be quite natural for them, then, to acknowledge that the information
about which hypothesis possess the highest degree of explanatoriness is as significant a constraint on our evolving
probability function as the raw evidence itself. We will need to take this new constraint into account if we are to
have any hope of successfully applying Bayes Theorem to the case at hand.
Just as we specify a range of significant outcomes for the raw evidence, e1 and e2, we specify a range of
outcomes for the new constraint “information about explanatory superiority”:
d1: H1 is the best explanation—it fits the best with background theory/causal principles once the evidence comes in
d2: H2 is the best explanation—it fits the best with background theory/causal principles once the evidence comes in.
We will then conditionalize on one of e1&d1, e1&d2, e2&d1, and e2&d2 (depending on what happens) when we
update our probability function from p to p'. So, for example, if e1&d1 turns out to be true, p'(e1&d1) = 1, p'(H1) =
p(H1/e1&d1), and p'(H2) = p(H2/e1&d1). We can use Bayes’ theorem to calculate these latter two values:
p'(H1) = p(H1/e1&d1) = p(e1&d1/H1)p(H1)
p(e1&d1)
p'(H2) = p(H2/e1&d1) = p(e1&d1/H2)p(H2)
p(e1&d1)
Now we see that there is no longer a difficulty with maintaining explanationist inclinations and respecting Bayes’
Theorem for the problematic sort of case: p'(H1) > p'(H2), since p(H1) = p(H2), and p(e1&d1/H1) > p(e1&d1/H2)
(explanationists will think that it is more likely that e1 happens and H1 turns out to be the best explanation if H1 is
in fact true, than if H2 is in fact true).
Neither van Fraassen’s argument, nor more general probabilistic considerations show that commitment to
explanationism results in probabilistic incoherence. Instead, the most significant moral to be drawn from reflections
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on Peter’s foibles and attendant worries about the compatibility of explanationism and Bayesianism is that
explanationists need to fill in the details of their position by specifying when explanatoriness allows us to boost our
degree of belief and how this boost should be incorporated into our changing belief state.
1
1. IBE has been thought to be potentially helpful in helping us combat Cartesian skepticism: our ordinary,
common-sense conception of the world is rational and justified—despite the existence of radical alternatives which
are equally compatible with our evidence—because the common sense conception offers a better explanation for the
play and patterns of our sensory appearances. Bertrand Russell, Jonathan Vogel, Paul Moser, Alan Goldman, Frank
Jackson, J. L. Mackie, Jonathan Bennett, James Cornman, and Michael Slote all pursue some version of this
approach. Gilbert Harman designates IBE as the most fundamental form of inductive inference and, along similar
lines, Laurence Bonjour depicts explanatory connections between beliefs as a major contributor to a belief system’s
coherence, and hence as a central factor in his accounts of justification. He also appeals to inference to the best
explanation in his early efforts to defend the idea that coherence in the sense he describes is relevant to a belief
system’s likelihood of truth. Abductive arguments for realism, whereby the approximate truth of our scientific
theories is offered as the best explanation for their empirical success, have been advocated with varying degrees of
refinement by J. J. C. Smart, Hillary Putnam, Richard Boyd and, most recently, Philip Kitcher. In his scathing
critique of contemporary analytic metaphysics, Bas van Fraassen protrays IBE as a harmful enabler of this
problematic philosophical enterprise, citing David Lewis’ defense of possible worlds, and David Armstrong’s
defense of universals as examples. IBE has played a longstanding role in the philosophy of religion on account of
its operation in both classic and newfangled arguments from design (“intelligent design theory”: the latest offering
from members of the creationist camp).
2
Acceptance of IBE might seem essential to scientific realism on account of the role abductive arguments have
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often played in the defense of scientific realism, though Miller (1987) and Ganson(2001) offer approaches to
defending realism which don’t depend on such arguments.
3
Developing a Dutch book is a way of showing how certain epistemic decisions are irrational or incoherent—in
violation of the axioms of probability. An implicit assumption in such an approach is that any time a bookie could
use an awareness of your principles for probability assignment to sell you a series of individually fair bets which
would in combination represent a sure loss for you, regardless of outcomes, your assignments (and the principles
dictating such assignments) are irrational. Given that a fair price for a bet on A which pays W if you win (0 if you
don't) is p(A)W, a bookie can determine which bets you would consider fair on the basis of your degrees of belief
and rules for degree of belief revision. If he can then offer you a series of individually fair bets which would lead to
a certain loss, your belief state as a whole must contain a structural inconsistency.
4
Here is a more thorough version of the Bayesian Peter story:
Peter is confronted with an alien die with a bias of any one of X(1), X(2)...X(10), he knows not which (again, a bias
of X(n) indicates that the probability of ace on any single proper toss is n/10). A friend comes along, and proposes
some bets:
Given proposition E: the first four tosses of the alien die show ace
and proposition H: the fifth toss shows ace
Bet I pays $10,000 if E is true and H is false
Bet II pays $1300 if E is false
Bet III pays $300 if E is true.
At the beginning of the die tossing, when Peter has the same initial probabilities as his friend, they both
evaluate the fair cost of each bet in accordance with the laws of the probability calculus.
The initial probability that E is true = the average of (.1)4, (.2)4,...,(1)4 = .25333 (Van Fraassen's ending with (.9)4 in
the text is just a typographical error: the initial probability for E which he states is correct.)
Initial probability that E is false = 1-.25333= .74667
Initial probability that E is true and H is false = the average of (.1) 4(.9),...,(.9)4(.1), 0 =.032505
The fair cost of a bet that pays x if y is xp(y) so:
Bet I costs $325.05
Bet II costs $970.67
Bet III costs $76.00
Peter buys all three bets from his friend ($1371.72). We see that if all four tosses do not come up ace, Peter
loses bets I and III, and wins bet II. In this case, Peter's losses amount to 1,371.71 - 1,300= 71.72.
Van Fraassen asks us to assume that E turns out true. Peter wins III and loses II, so he is paid $300. So far
he has a net loss of $1,071.72. At this point, information from an earlier part of the story which I haven't included
yet becomes relevant. Before running into his friend, Bayesian Peter was confronted by an IBE preacher who
convinced him that the explanatory power of a hypothesis allows him to raise his credence. Peter came to believe
that posterior probability can be fixed by prior probability, outcomes, and explanatory success (see van Fraassen
1989, p. 166).
In light of this belief and the string of four ace tosses, which is best explained by a hypothesis of high bias,
Peter raises his probability for H being true from .87 (the result obtained from a routine application of Bayes’
Theorem) to .9. Peter sells bet I to his friend for $1,000 (the payoff is $10,000, and the probability that H is false is
.1).
5
Before he embarks on his final transaction and sells bet I for $1,000, he has spent a total of only $999.97, not
$1,071.72 as in the original version of the parable.
6
To see that the Bayesian Peter bets do have this structure, note that y = 10,000,
7
More generally, I and II together are essentially equivalent to a bet on -H given E which costs xy and pays y. The
total cost of these bets is yp(-H & E) + xyP(-E) = yP(-H/E)P(E) xyP(-E) =
Though not his primary criticism of van Frassen’s argument, Douven (1999) draws a similar conclusion:
“Presumably not even the staunchest defender of IBE would want to hold that IBE is applicable in every context.
Indeed, it is exactly the sort of context in which van Fraassen puts the discussion that seems not to license an
inference to the best explanation...IBE is a contextual principle in that it draws heavily upon background knowledge
for its application, e.g., in order to judge which hypothessis among a number of rivals is the best explanation for the
8
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data gathered. However, in van Fraassen’s model there is no such knowledge to invoke...” (Douven 1999)
9
Kvanig (1994) and Leeds (1994) also criticize van Fraassen’s argument by noting that van Fraassen needs to say
more to defend the notion that diachronic incoherence is indicative of a failure of epistemic rationality, though there
are flaws in the way they develop this objection. Kvanig claims that diachronic Dutch books, unlike sychronic ones,
illegitimately rely on the bookie’s possessing “special information”, namely knowledge of the future. According to
Kvangiv, being susceptible to a certain loss when dealing with a prescient bookie is not epistemically telling: after
all, we would all be susceptible to certain loss if we accepted bets from omniscient bookies. This observation
hardly seem to capture why the epistemic implications of synchronic and diachronic incoherence may be
fundamentally different: the bookie in a diachronic Dutch book has no uncanny knowledge of the future; he simply
knows your current degrees of belief and rules for degree of belief revision, and hence knows which bets you’ll
regard as fair over time.
Leeds casts doubt on the epistemic implications of probabilistic incoherence by noting: “...if we reject the
operationalist idea–prevalent for so long in both psychology and economics–that we need to define the notion of
degree of belief in terms of betting behavior, then we will have no reason to think there are such tight conceptual
connections between betting and believing as Lewis’ argument requires.” (Leeds 1994, p. 211) We can adequately
justify taking Dutch books (at least synchronic ones) as epistemically relevant without presupposing that degree of
belief needs to be defined in terms of betting behavior, however. As Howson and Urbach (1993) suggest, a belief in
A to degree p(A) implies that the agent would at least regard a bet on A costing p(A)w with payoff w as fair, i.e. as
offering zero advantage to either side of the bet, given the way the world appears to the agent. If the agent
simultaneously regards particular individual bets as fair (offering neither advantage nor disadvantage), the sum of
such bets should also offer neither advantage nor disadvantage. In the case of Dutch books, however, it turns out
that the sum, by guaranteeing a net loss if purchased, offers a definite disadvantage and hence cannot be fair. If a
Dutch book has been developed from bets the agent simultaneously regards as individually fair, we see that there
must have been a clash in her original simultaneous degree of belief assignments, a lack of coherence amongst those
values which dictate which bets she counts as fair.
In light of the Dutch Book theorem, this line of thought provides an excellent justification for the view that
our simultaneous degree of belief assignments have to conform to the axioms of probability in order to be rational.
In the discussion which follows I don’t presuppose any specific account of what more, of epistemic significance,
there is to good explanation. All that we are concerned with here is the general view that explanatoriness is
sometimes relevant to degree of credence, even when the explanatory superiority of a favored hypothesis is
grounded in more than its predictive success. The reader should keep in mind, however, that explanationists have
contrasting conceptions of what can make explanatoriness epistemically relevant.
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Bibliography
Ben-Menahem, Y., “The Inference to the Best Explanation,” Erkenntnis 33 (199).
Day, T., and Kincaid, H., “Putting Inference to the Best Explanation in its Place,” Synthese 98 (1994).
Douven, I., “Inference to the Best Explanation Made Coherent,” Philosophy of Science 66 (1999).
Ganson, D., The Explanationist Defense of Scientific Realism (Garland, 2001).
Harman, G., Reasoning, Meaning, and Mind (Oxford, 1999).
Howson, C., and Urbach, P., Scientific Reasoning: The Bayesian Approach (Open Court, 1993).
Kvanig, J., “A Critique of van Frassen’s Voluntaristic Epistemology,” Synthese 98 (1994).
Leeds, S., “Constructive Empiricism”, Synthese 101 (1994).
Miller, R., Fact and Method (Princeton, 1987).
Okasha, S., “Van Fraassen’s Critique of Inference to the Best Explanation,” Stud. Hist. Phil., Vol 31, No. 4 (2000).
van Fraassen, B., “Belief and the Will”, Journal of Philosophy 81 (1984).
van Fraassen, B., The Empirical Stance, (Yale, 2002).
van Fraassen, B., Laws and Symmetry (Oxford, 1989).
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