The Probabilistic No Miracles Argument

Jan Sprenger∗

July 23, 2015

Abstract

This paper develops a probabilistic reconstruction of the No Miracles Argu-
ment (NMA) in the debate between scientific realists and anti-realists. The goal
of the paper is to clarify and to sharpen the NMA by means of a probabilistic
formalization. In particular, we demonstrate that the persuasive force of the NMA
depends on the particular disciplinary context where it is applied, and the stability
of theories in that discipline. Assessments and critiques of “the” NMA, without
reference to a particular context, are misleading and should be relinquished. This
result has repercussions for recent anti-realist arguments, such as the claim that
the NMA commits the base rate fallacy (Howson, 2000; Magnus and Callender,
2004). It also helps to explain the persistent disagreement between realists and
anti-realists.

∗Contact information: Tilburg Center for Logic, Ethics and Philosophy of Science (TiLPS), Tilburg
University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands. Email: j.sprenger@uvt.nl. Webpage:
www.laeuferpaar.de

1



1 Introduction

The debate between scientific realists and anti-realists is one of the classics of philoso-

phy of science, comparable to a soccer match between Brazil and Argentina. Realism

comes in different varieties, e.g., metaphysical, semantic and epistemological realism

(see Chakravartty, 2011, for a survey). In this paper, we focus on the epistemological

thesis that we are justified to believe in the truth of our best scientific theories, and that

they constitute knowledge of the external world (Boyd, 1983; Psillos, 1999, 2009). In

this view, the existence of a mind-independent world (metaphysical realism) and the

reference of theoretical terms to mind-independent entities (semantic realism) is usu-

ally presupposed—the real question concerns the epistemic status of our best scientific

theories.

A major player in this debate is the No Miracles Argument (NMA). It contends

that the truth of our best scientific theories is the only hypothesis that does not make

the astonishing predictive, retrodictive and explanatory success of science a mystery

(Putnam, 1975, 73). If our best scientific theories did not correctly describe the world,

why should we expect them to be successful? On the other hand, since the truth of our

best theories is an excellent explanation of their success, we should accept the realist

hypothesis: our best scientific theories are true and constitute knowledge of the world.

It is not entirely clear whether the NMA is an empirical or a super-empirical ar-

gument. As an argument from past and present success of our best scientific theories

to their truth, it involves two major steps: the step from observed success to rational

belief in empirical adequacy, and the step from rational belief in empirical adequacy

to rational belief in truth (see Figure 1). The first of them is an empirical inference, the

second most probably not: ordinary empirical evidence cannot distinguish between

different theoretical structures that yield the same observable consequences.

Much philosophical discussion has been devoted to the second step of the NMA

(e.g., Psillos, 1999; Lipton, 2004; Stanford, 2006), which seems in greater need of a

philosophical defense. After all, the realist has to address the problems of underde-

termination of theory by evidence and the existence of unconceived alternatives. But

also the first step of the NMA is far from trivial, and strengthening it against criticism

is vital for the scientific realist. For instance, Laudan (1981) has argued that there have

been lots of successful, but non-referring scientific theories. If Laudan were right, then

the entire NMA would break down, even if objections to the second step could be

countered successfully.

2



Figure 1: The structure of the NMA as a two-step argument from the empirical success of T
to its truth. In our model, we conceptualize the NMA as an argument for the first inference in
this figure, that is, for an inference from empirical success of T to its empirical adequacy.

Arguments against the first step of the NMA do not only threaten full scientific

realists, but also structural realists (Worrall, 1989) and some varieties of anti-realism

that stick out their neck. One of them is Bas van Fraassen’s constructive empiricism
(van Fraassen, 1980; Monton and Mohler, 2012). Proponents of this view deny that we

have reasons to believe that our best scientific theories are literally true. However, they

affirm that we are justified to believe in the observable parts of our best theories. Thus
they are also affected by criticism and defense of the first step of the NMA.

Hence, the first step of the NMA does not draw a sharp divide between realists

and anti-realists. Rather, the debate takes place between those who derive epistemic

commitments from the success of science, and those who deny them. For convenience,

we stick to the traditional terminology and refer to the first group as “realists” and to

the second group as “anti-realists”.

The paper aims to show the realist and anti-realist standpoints can be reconciled

with probabilistic rationality, and Bayesian rationality in particular. The paper assumes

that rationally believed propositions should also be probable (for discussion of this

thesis, see e.g., Leitgeb, 2014). We set up a simple probabilistic model of the NMA and

use it to underline a recent criticism by Colin Howson (2000, 2013) that the NMA falls

prey to the base rate fallacy (Section 2). We then develop a refined probabilistic model

of the NMA where this criticism is addressed, and where the stability of scientific

theories and the existence of relevant alternatives are accommodated (Section 3). By

assigning a crucial role to the nature and history of the relevant scientific discipline,

3



this model allows for a more nuanced assessment of the NMA. The paper concludes

with a short discussion of the scope of the NMA, and a plea for context-sensitive

assessment of its validity (Section 4).

2 A Probabilistic No Miracles Argument

The NMA is sometimes understood as a general argument for believing in the truth of
our best scientific theories. In this paper, we restrict it to a particular scientific theory
T which is predictively and explanatorily successful in a certain scientific domain.
Since we only investigate arguments for the empirical adequacy of T, we introduce a
propositional variable H—the hypothesis that T is empirically adequate. See Figure 2
for a simple Bayesian network representation of the dependence between H and the
propositional variable S that represents the empirical success of T.

H S

Figure 2: The Bayesian Network representation of the impact of H—the empirical adequacy
of theory T—on the empirical success of T, denoted by S.

Expressed as a probabilistic argument, the simple NMA then runs as follows: S is
much more probable if T is empirically adequate than if it is not:

p(S|H) � p(S|¬H)

From Bayes’ Theorem, we can then infer

p(H|S) > p(H)

In other words, S confirms H: our degree of belief in the empirical adequacy of T
is increased if T is successful.1 Here, we have interpreted the above probabilities as
rational degrees of beliefs, and the NMA as proving an increase in degree of belief.

This subjective Bayesian interpretation of the NMA strikes us as natural. However, the

validity of the argument is not affected if an objective interpretation of probability is

adopted instead. This is the reason why we refer to the probabilistic rather than the

1See Hartmann and Sprenger (2010) for an introduction to Bayesian epistemology with applications
to topics in philosophy of science, such as explicating degree of confirmation.

4



Bayesian NMA. We leave the preferred interpretation of probability to the reader’s

taste and do not consider this issue further.

To the above argument, anti-realists object that the inequality p(H|S) > p(H) falls
short of establishing the realist claim. We are primarily interested in whether H is
sufficiently probable given S, not in whether S raises the probability of H. After all, the
increase in probability could be negligibly small. The result p(H|S) > p(H) does not
establish that p(H|S) is beyond a critical threshold, e.g., a probability of 1/2.

More specifically, it has been argued that the NMA commits the base rate fallacy
(Howson, 2000; Magnus and Callender, 2004). This is an unwarranted type of inference

that frequently occurs in medicine. Consider a highly sensitive medical test which

yields a positive result. On the other hand, the medical condition in question is very

rare, that is, the base rate of the disease is very low. In such a case, the posterior

probability of the patient having the disease, given the test, will still be quite low.

Nonetheless, medical practitioners tend to disregard the low base rate and to infer

that the patient really has the disease in question (e.g., Goodman, 1999).

This objection can be elucidated by a brief inspection of Bayes’ Theorem. Our

quantity of interest is the posterior probability p(H|S), our confidence in H given S.
This quantity can be written as

p(H|S) =
p(H) p(S|H)

p(S)

=

(
1 +

1 − p(H)
p(H)

p(S|¬H)
p(S|H)

)−1
(1)

which shows that p(H|S) is not only an increasing function in p(S|H) and a decreasing
function in p(S|¬H): its value crucially depends on the base rate or prior plausibility
of H, p(H) (=the probability that T is empirically adequate).

Anti-realists claim that NMA is built on a base rate fallacy: from the high value

of p(S|H) (“the empirical adequacy of T explains its success”) and the low value
of p(S|¬H) (“success of T would be a miracle if T were not empirically adequate”),
rational belief in H (“T is empirically adequate”) is inferred. The probabilistic model of
the NMA demonstrates that we need additional assumptions about p(H) to warrant
this inference. In the absence of such assumptions, the NMA does not entitle us to

accept T as empirically adequate.
What do these considerations show? First, they expose that the NMA, recon-

5



structed as a probabilistic inference to the posterior probability of H, is essentially
subjective. After all, any weight of evidence in favor of H can be counterbalanced by a
sufficiently skeptical prior, that is, a sufficiently low value assigned to p(H). The anti-
realist contends that the realist needs to provide convincing reasons why p(H) should
not be arbitrarily close to zero, and that such reasons will typically presuppose realist

inclinations. This is a problem for those realists who claim that the NMA is an objective
argument in favor of scientific realism. Howson (2013, 211) concludes that due to the

dependence on unconstrained prior degrees of belief, the NMA is, “as a supposedly

objective argument, [. . . ] dead in the water”. See also Howson (2000, ch. 3), Lipton

(2004, 196–198), and Chakravartty (2011).

Second, the anti-realist may argue that empirical adequacy is not required for pre-

dictive success. Every now and then, science undergoes radical discontinuities: it is

discovered that central terms in scientific theories do not refer, that theories fail to

apply outside a particular domain, etc. The theories which supersede them are empir-

ically distinct. Why should our currently best theory Tn = T not suffer the same fate as
it predecessors T1, . . . , Tn−1 which proved to be empirically inadequate although they
once were the best scientific theory? This is basically Laudan’s argument from Pes-

simistic Meta-Induction (PMI): “I believe that for every highly successful theory in the

past of science which we now believe to be a genuinely referring theory, one could find

half a dozen successful theories which we now regard as substantially non-referring”

(Laudan, 1981, 35). See Tulodziecki (2015) for a recent case study.

PMI affects the values of p(S|¬H) and p(H) as follows: On the one hand, history
teaches us that there have often been false theories that explained the data well (and

were superseded later). In other words, empirically inadequate theories can be highly

successful and p(S|¬H) need not be that low. Second, the fact that T1, . . . , Tn−1 stand
refuted, plus possible continuities and structural similarities between those theories

and Tn = T, suggest, by virtue of an inductive inference, that T may ultimately suffer
the same fate, lowering the probability that T is empirically adequate.

Let us check these arguments in a numerical analysis of the probabilistic NMA.

Conceding a bit to the realist, we set s := p(S|H) = 1: if theory T is empirically
adequate, then it is also successful.2 Furthermore, define s′ := p(S|¬H) and let h :=
p(H) be the prior probability of H. We now ask the question: for which values of

2Note that empirical adequacy does not guarantee predictive success: e.g., if T is a low-level theory
with many parameters to be estimated simultaneously, T may make less accurate predictions than an
empirically inadequate, but simpler theory (Forster and Sober, 1994).

6



s′ and h is the posterior probability of H, p(H|S), greater than 1/2? That is, when
would it be more plausible to believe that T is empirically adequate than to deny it?
This condition is arguably a minimal requirement for the claim that the success of T
entitles us to rational belief in its empirical adequacy.

By using Bayes’ Theorem, we can easily calculate when the inequality p(H|S) >
1/2 is satisfied. Equation (1) brings us to the inequality

1
2
<

(
1 + s′

1 − h
h

)−1
which can be written as

s′ <
h

1 − h
(2)

See Figure 3 for a graphical illustration.

Figure 3: The scope of the No Miracles Argument, represented graphically. p(H|S) > 1/2 is
the case in the white area below the line.

However, inequality (2) is not easy to satisfy. As mentioned above, false theories

and models often make accurate predictions and perform well on other cognitive val-

ues (see Frigg and Hartmann, 2012, for an overview). Classical examples that are

still used today involve Newtonian mechanics, the Lotka-Volterra model from popu-

lation biology (e.g., Weisberg, 2007) and Rational Choice Theory. These theories may

be false, but they are definitely useful and successful. Given the frequency of such

7



theories in science, we may estimate s′ = 1/4—the precise value matters less than the
order of magnitude. To satsify inequality (2) and to make the NMA work, we would

then require that p(H) ∈ [1/3, 1]! In other words, the NMA only flies for theories
with a substantial prior probability of being empirically adequate. What is more, for

a “mildly skeptical prior” such as p(H) = 0.05, the value of s′ would have to be in
the range [0, 0.05]. This amounts to making the assumption that only the empirical

adequacy (or truth) of a scientific theory can explain its success. But this is essentially

a realist premise which the anti-realist would refuse to accept. She could point to the

existence of unconceived alternatives (Stanford, 2006, ch. 6), the explanatory successes

of false theories, etc. In other words: the simple probabilistic model of the NMA

demonstrates that (1) to the extent that the NMA is valid, its premises presuppose

realist inclinations; (2) to the extent that the NMA builds on premises that are neutral

between the realist and the anti-realist, it fails to be valid.

Are things thus hopeless for the realist who wants to convince the anti-realist

that the NMA is a good argument? Does “all realistic hope of resuscitating the [no

miracles] argument [fail]”, as Howson (2013, 211) writes? Perhaps not necessarily

so. The next section develops another, more sophisticated probabilistic model whose

results are more friendly toward the realist position.

3 A New Model of the No Miracles Argument

So far, the probabilistic NMA only took the predictive and explanatory success of T
as evidence for the realist position. But perhaps, different kinds of evidence bear on

the argument, too. For example, Ludwig Fahrbach (2009, 2011) has argued that the

stability of scientific theories in recent decades favors scientific realism. In this section,

we show how such arguments could be part of a probabilistic NMA. We do not want

to take a stand on the historical correctness of Fahrbach’s observations: rather, we

would like to demonstrate how such observations can in principle affect the NMA.
Fahrbach’s argument is based on scientometric data. He observes an exponential

growth of scientific activity, with a doubling of scientific output every 20 years (Mead-

ows, 1974). He also notes that at least 80% of all scientific work has been done since

the year 1950 and observes that our best scientific theories (e.g., the periodic table of

elements, optical and acoustic theories, the theory of evolution, etc.) were stable dur-

ing that period of time. That is, they did not undergo rejection or major conceptual

change. Laudan’s examples in favor of PMI, on the other hand, all stem from the

8



early periods of science: the caloric theory of heat, the ether theory in physics, or the

humoral theory in medicine.

For giving a fair assessment of PMI, we have to take into account the amount of

scientific work done in a particular period. This implies, for example, that the period

1800–1820 should receive much less weight than the period 1950–1970. According to

Fahrbach, PMI fails because most “theory changes occurred during the time of the

first 5% of all scientific work ever done by scientists” (Fahrbach, 2011, 149). If PMI

were valid, we should have observed more substantial theory changes or scientific

revolutions in the recent past. However, although the theories of modern science often

encounter difficulties, revolutionary turnovers do not (or only very rarely) happen.

According to Fahrbach, PMI stands refuted—or at the very least, it is not more rational

than optimistic meta-induction.
The factual correctness of Fahrbach’s observations may be disputed, and his model

is certainly very simplified. Yet, it deserves to be taken seriously, particularly with

respect to the implications for the NMA. In this paper, we explore if observations of

long-term stability expand the set of circumstances where the NMA holds. To this

end, we refine our probabilistic model of the NMA.

As before, the propositional variable H expresses the empirical adequacy of theory
T, and S denotes the predictive, retrodictive and explanatory success of T. Similar to
Dawid et al. (2015), we introduce a integer-valued random variable A that expresses
the number of satisfactory alternatives to T, including unconceived theories. In the
individuation of alternatives, we stick with the Dawid et al. paper: that is, we demand

that alternative theories satisfy a set of (context-dependent) theoretical constraints C,
are consistent with the currently available data D, and give distinguishable predic-
tions for the outcome of some set E of future experiments. In line with our focus
on empirical adequacy rather than truth, we do not distinguish between empirically

equivalent theories with different theoretical structures. Finally, major theory change

in the domain of T is denoted by variable C, and absence of change and theoretical
stability by ¬C. “Theory change” is understood in a broad sense, including scenarios
where rivalling theories emerge and end up co-existing with T.

The dependency between these four propositional variables—A, C, H and S—is
given by the Bayesian network in Figure 4. S, the success of theory T, only depends on
the empirical adequacy of T, that is, on H. The probability of H depends on the num-
ber of distinct alternatives that are also consistent with the current data, etc. Finally, C,
the probability of observing substantial theory change, depends on S and A: the em-

9



pirical success of T and the number of available alternatives to T that scientists might
develop. H affects C only via A. All this is captured in the following assumption:

A H S

C

Figure 4: The Bayesian Network representation of the relation between variables A (the num-
ber of alternatives to T), H (empirical adequacy of theory T), S (the success of T) and C (major
theory change).

A0 The variables A, C, H and S satisfy the conditional independencies in the Bayesian
Network of Figure 4.

To rule out preservation of a theory by a of series degenerative accommodating

moves, the variable C should be evaluated over a longer period (e.g., 30–50 years). We
now define a number of real-valued variables in order to facilitate calculations:

• Denote by aj := p(A = j) the probability that there are exactly j (not necessarily
discovered) alternatives to T that satisfy the theoretical constraints C, are consis-
tent with current data D and give definite predictions for future experiments E,
etc.3

• Denote by hj := p(H|A = j) the probability that T is empirically adequate if
there are exactly j alternatives to T.

• As before, denote by s := p(S|H) and s′ := p(S|¬H) the probability that T is
successful if it is (not) empirically adequate.

• Denote by cj := p(¬C|A = j, S) the probability that no substantial theory change
occurs if T is successful and there are exactly j alternatives for T.

Suppose that we now observe ¬C (no substantial theory change has occurred in
the last decades) and S (theory T is successful). The Bayesian network structure allows

3It is assumed that there are only finitely many alternative theories that satisfy these quite restrictive
specifications. However, the number of alternatives may be arbitrarily large. See Dawid et al. (2015) for
more discussion of that assumption.

10



for a simple calculation of the posterior probability of H:

p(¬CSH) = ∑
A

p(A)p(¬C|AS)p(S|H)p(H|A)

=
∞

∑
j=0

aj cj s hj

p(¬CS) = ∑
A,H

p(A)p(¬C|AS)p(S|H)p(H|A)

= ∑
A

p(A)p(¬C|AS)p(S|H)p(H|A) + ∑
A

p(A)p(¬C|AS)p(S|¬H)p(¬H|A)

=
∞

∑
j=0

aj cj (s hj + s
′(1 − hj))

With the help of Bayes’ Theorem, these equations allows us to calculate the posterior

probability of H conditional on C and S. That is, how probable is H given that T is
successful and that we have observed no major theoretical change in recent decades?

p(H|¬CS) =
p(¬CSH)
p(¬CS)

=
∑∞j=0 aj cj s hj

∑∞j=0 aj cj (s hj + s′(1 − hj))
(3)

We now make some assumptions on the values of these quantities.

A1 If T is empirically adequate then it will be successful in the long run: s = p(S|H) =
1.4

A2 The empirical adequacy of T is no more or less probable than the empirical ade-
quacy of an alternative which satisfies the same set of theoretical and empirical

constraints: hj := p(H|Aj) = 1/(j + 1). Scientists are indifferent between theo-
ries which satisfy the same set of theoretical and empirical constraints.

A3 The more satisfactory alternatives are in principle accessible to the scientists, the

less likely is an extended period of theoretical stability. In other words, cj :=
p(¬C|A = j, S) is a decreasing function of j. For convenience, we choose cj =
1/(j + 1).

While A2 is a default assumption made for expositional ease, A3 is more substantive:

4The formulation p(S|H) = 1 − e is more cautious, but this not change the qualitative results and
only complicates the calculations.

11



ceteris paribus, the probability of long-term theoretical stability decreases the more

satisfactory alternatives for a successful theory exist. This qualitative claim is very

intuitive. More contentious is the precise rate of decline. That’s why we will relax the

peculiar assignment of the cj later on in the paper.

A4 Assume that T is our currently best theory and we happen to find a satisfactory
alternative T′. Then, the probability of finding yet another alternative T′′ is the
same as the probability of finding T′ in the first place. Formally:

p(A > j|A ≥ j) = p(A > j + 1|A ≥ j + 1) ∀j ≥ 0. (4)

In other words, equation (4) expresses the idea that finding an alternative does

not, in itself, raise or lower the probability of finding another alternative.

Note that A0–A4 are equally plausible for the realist and the anti-realist. In other

words, no realist bias has been incorporated into the assumptions. We can now show

the following proposition (proof in the appendix):

Proposition 1: If a0 > 0, then A4 is equivalent to the requirement aj := a0 · (1 − a0)j.

Together with this proposition, A0–A4 allow us to rewrite equation (3) as follows:

p(H|¬CS) =
∑∞j=0(1 − a0)j

1
(j+1)2

∑∞j=0(1 − a0)j
1−s′j
(j+1)2

(5)

With the help of this formula, we can now rehearse the NMA once more and

determine its scope, that is, those parameter values where p(H|¬CS) > 1/2. The two
relevant parameters are a0, the probability that there are no satisfactory alternatives to
T, and s′, the probability that T is successful if it is not empirically adequate. Since
an analytical solution of equation (5) is not feasible, we conduct a numerical analysis.

Results are plotted in Figure 5.

These results are very different from the ones in Section 2. With the hyperplane

z = 0.5 dividing the graph into a region where T may be accepted and a region where
this is not the case, we see that the scope of the NMA has increased substantially

compared to Figure 3. For instance, a0 > 0.1 suffices for a posterior probability greater
than 1/2, even if the value of s′ is very high. This is a striking difference to the previous

12



Figure 5: The scope of the No Miracles Argument in the revised formulation. The posterior
probability of H, p(H|¬CS), is plotted as a function of (1) the prior probability that T is
empirically adequate (a0); (2) the probability that T is successful if T is false (s′ = p(S|¬H)).
The hyperplane z = 1/2 is inserted in order to show for which parameter values p(H|¬CS) is
greater than 1/2.

analysis where way more optimistic values had to be assumed in order to make the

NMA work.

So far, the analysis has been conducted in terms of absolute confirmation, that is, the
posterior probability of H. We now complement it by an analysis in terms of relative
or incremental confirmation. That is, we calculate the degree of confirmation that ¬CS
exerts on H. We use four different measures to show the robustness of our results.
First, we calculate the log-likelihood measure l(H, E) = log2 p(E|H)/ p(E|¬H) which
has a good reputation in formal theories of evidence (e.g., Hacking, 1965; Fitelson,

2001) and a firm standing in scientific practice (e.g., Royall, 1997; Good, 2009). To

apply it to the present case, we calculate

p(¬CS|H) =
p(¬CSH)

p(H)
=

∑A p(A)p(¬C|AS)p(S|H)p(H|A)
∑A p(A) p(H|A)

=
∑∞j=0 aj cj s hj

∑∞j=0 aj hj

=
∑∞j=0(1 − a0)j

1
(1+j)2

∑∞j=0(1 − a0)j
1

1+j

13



p(¬CS|¬H) =
p(¬CS¬H)

p(¬H)
=

∑A p(A)p(¬C|AS)p(S|¬H)p(¬H|A)
∑A p(A) p(¬H|A)

=
∑∞j=0 aj cj s

′ (1 − hj)
∑∞j=0 aj (1 − hj)

=
∑∞j=0(1 − a0)j

s′j
(1+j)2

∑∞j=0(1 − a0)j
j

1+j

l(H, E) is a confirmation measure that describes the discriminative power of the
evidence with respect to the realist and the anti-realist hypothesis. It is relatively

insensitive to prior probabilities. Other measures aim at quantifying the increase in

degree of belief from p(H) to p(H|E). Typical representatives of that class are the log-
ratio measure r(H, E), the difference measure d(H, E) and the Crupi-Tentori measure
z(H, E). The definitions of all measures are given below (see also Crupi, 2013), and
their values follow straightforwardly from the above calculations.

l(H, E) = log2
p(E|H)

p(E|¬H)
r(H, E) = log2

p(H|E)
p(H)

d(H, E) = p(H|E)− p(H) z(H, E) =
p(H|E)− p(H)

1 − p(H)
(if p(H|E) > p(H))

To deliver a comprehensive picture and to show the robustness of our claims, we

calculate the degree of confirmation for all four confirmation measures. All measures

are normalized such that a value of zero corresponds to neither confirmation nor

disconfirmation. In addition, d(H, E) and z(H, E) only take values between -1 and 1.
Figure 6 plots the degree of confirmation as a function of the value of s′, for three

different values of a0, namely 0.01, 0.05 and 0.1. As visible from the graph, the degree
of confirmation is substantial for all four measures. In particular, it is robust vis-à-

vis the values of a0 and s′, contrary to the anti-realist argument from Section 2. For
example, if s′ is quite small, as it will often be the case in practice, then l(H,¬CS)
comes close to 10, which corresponds to a ratio of more than 1,000 between p(¬CS|H)
and p(¬CS|¬H)! This finding accounts for the realist intuition that the stability of
scientific theories over time, together with their empirical success, is strong evidence

for their empirical adequacy.

Finally, we conduct a robustness analysis regarding A3. Arguably, the function

cj := p(¬C|A = j, S) = 1/(j + 1) suggests that scientists are quite ready to give

14



Figure 6: The degree of confirmation l(H,¬CS), r(H,¬CS), d(H,¬CS) and z(H,¬CS), for
three different values of a0. Full line: a0 = 0.01. Dashed line: a0 = 0.05. Dot-dashed line:
a0 = 0.1

up on their currently best theory in favor of a good alternative. But as many have

philosophers and historians of science have argued (e.g., Kuhn, 1977), scientists may

be more conservative and prefer the traditional framework, even if good alternatives

exist. Therefore we also analyze a different choice of the cj, namely cj := e
−12
(

j
α

)2
,

where cj falls more gently in j. This leads to the following expression for the posterior
probability of H:

p(H|¬CS) =
∑∞j=0(1 − a0)j

1
j+1 e

−12
(

j
α

)2

∑∞j=0(1 − a0)j
1−s′j
j+1 e

−12
(

j
α

)2
The graph of p(H|¬CS), as a function of a0 and s′, is presented in Figure 7. We have set
α = 4, which corresponds to a more gradual decline of cj. Yet, the results match those
from Figure 5: the scope of the NMA is much larger than in the simple version of the

probabilistic NMA. Hence, our findings seem to be robust toward different choices of

cj. To achieve a result similar to the one for the primitive NMA (Figure 3), one would
have to choose α = 8, which will only be the case if scientists are really reluctant to

reject the currently best theory. In such circumstances, stability is the default state of

15



a discipline and will not strongly support the NMA.

Figure 7: The scope of the No Miracles Argument in the revised formulation, with cj :=

e−
1
2 (

x
α )

2
. The posterior probability of H, p(H|¬CS), is plotted as a function of a0 and s′, like in

Figure 5, and contrasted with the hyperplane z = 1/2.

All in all, our model shows that a probabilistic NMA need not be doomed. Its va-

lidity depends crucially on the properties of the disciplinary context where it operates

in. This involves the existence of satisfactory alternatives and whether or not the dis-

cipline has been in a long period of theoretical stability. Of course, our model makes

simplifying assumptions, but unlike those in Section 2, they do not carry a realist bias.

This allows for a more nuanced and context-sensitive assessment of realist claims.

4 Conclusions

This paper has investigated scope and limits of the No Miracles Argument (NMA)

when formalized as a probabilistic argument. In the simple probabilistic model of the

NMA, we have confirmed Howson’s (2000, 2013) diagnosis that it does not hold water

as an objective argument: too much depends on the choice of the prior probability

p(H), assuming what is supposed to be shown. We have supported this diagnosis by
a detailed analysis of the probabilistic mechanics of NMA.

16



However, not all is lost for the NMA. We have developed a probabilistic model

of the NMA where additional evidence, such as the stability of scientific theories,

can be accommodated. This model also conceptualizes the possible alternatives to

the theory in question. Using this model, scientific realism can be defended with
much weaker assumptions than in the simple probabilistic version of the NMA. While

context-sensitive assumptions are required in this version of the NMA, their relative

weakness leaves open the possibility of a coherent, non-circular realist position in

philosophy of science. See also Dawid and Hartmann (2015).

We would like to stress that the context-sensitive nature of the NMA is not a vice,

but a virtue. True, context-sensitive elements undermine the persuasive force of the

NMA for those who do not already share realist inclinations. But when there is little

stability in our currently best theories, or when empirically successful alternatives

abound, there is nothing to feed realist intuitions in the first place. Hence it is natural

that contextual considerations determine when the NMA is valid and when it isn’t.

Our analysis in Section 3 has shown how much depends on the stability of scientific

theories in order to make the NMA fly. More research is needed into which areas of

science have been theoretically stable. It may be argued that scientific theories of

the recent past remained stable on the surface, e.g., in the basic assumptions they

make, but that central concepts tacitly underwent major meaning shifts. Evolutionary

biology is such a case in question: while basic principles such as the causal power

of natural selection to bring about evolutionary change have been unchanged, the

meaning of concepts such as natural selection, fitness, and selective environments has

been fiercely debated (Lloyd, 2014).

All this demonstrates that conceptualizations of the NMA as a general argument for
scientific realism are mistaken. Instead of reading NMA as a “wholesale argument”

for scientific realism that is valid across the board, we should understand it as a “retail

argument” (Magnus and Callender, 2004), that is, as an argument that may be strong

for some scientific theories and weak for others.

For philosophers like ourselves, who are not committed to a particular position

in the debate between realists and anti-realists, the probabilistic reconstruction of the

NMA offers the chance to understand the argumentative mechanics behind the realist

intuition, to better appreciate the context-dependency of the NMA, and to critically

evaluate the merits of realist and anti-realist standpoints. The realist argument is

based on contexts where theories are stable and there are few potential explanations

of empirical success. Such circumstances favor the NMA. The anti-realist objections are

17



grounded on those case studies where scientific theories have been volatile or one of

our assumptions A0–A4 is implausible. The probabilistic reconstruction of the NMA

can thus explain and guide the strategies that realists and anti-realists pursue when

defending their positions.

Appendix: Proof of Proposition 1

We begin with the “⇒” direction of the proposion. Assumption A4 is equivalent to
the following claim:

p(A > j|
∞∨

k=j

(A = k)) = p(A > j + 1|
∞∨

k=j+1

(A = k)) ∀j ≥ 0

which is again equivalent to

p(A = j|
∞∨

k=j

(A = k)) = p(A = j + 1|
∞∨

k=j+1

(A = k)) ∀j ≥ 0

This entails that for all j ≥ 0, we have

p(A = j)
p(
∨∞

k=j(A = k))
=

p(A = j + 1)
p(
∨∞

k=j+1(A = k))

This implies in turn

p(A = j + 1) = p(A = j)
p(
∨∞

k=j+1(A = k))
p(
∨∞

k=j(A = k))

= p(A = j)
1 − ∑jk=0 p(A = k)
1 − ∑j−1k=0 p(A = k)

By a simple induction proof, we can now show

p(A = n) = p(A = 0)

(
1 −

n−1
∑
k=0

p(A = k)

)
(6)

18



For n = 1, equation (6) follows immediately. Assuming that it holds for level n, we
then obtain

p(A = n + 1) = p(A = n)
1 − ∑nk=0 p(A = k)
1 − ∑n−1k=0 p(A = k)

= p(A = 0)

(
1 −

n−1
∑
k=0

p(A = k)

)
1 − ∑nk=0 p(A = k)
1 − ∑n−1k=0 p(A = k)

= p(A = 0)

(
1 −

n

∑
k=0

p(A = k)

)

where we have used the inductive premise in the second step. Finally, we use straight

induction once more to show that

p(A = n) = p(A = 0)(1 − p(A = 0))n (7)

where the case n = 0 is trivial and the inductive step n → n + 1 is proven as follows:

p(A = n + 1) = p(A = 0)

(
1 −

n

∑
k=0

p(A = k)

)

= p(A = 0)

(
1 −

n

∑
k=0

p(A = 0) (1 − p(A = 0))k
)

= p(A = 0)
(

1 − p(A = 0)
1 − (1 − p(A = 0))n+1

1 − (1 − p(A = 0))

)
= p(A = 0)(1 − (1 − (1 − p(A = 0))n+1))

= p(A = 0)(1 − p(A = 0))n+1

In the second line, we have applied the inductive premise to p(A = k), and in the third
line, we have used the well-known formula for the geometric series:

n

∑
k=0

qk =
1 − qn+1

1 − q
(8)

This shows (7) and completes the proof of the “⇒” part of the proposition. The ”⇐”
direction is a purely algebraic exercise: using equation (8) and the fact that ∑∞k=0 ak = 1,

19



we can easily calculate that

p(A ≥ j) = (1 − a0)j p(A ≥ j + 1) = (1 − a0)j+1

from which it follows that

p(A = j)
p(A ≥ j)

= a0 =
p(A = j + 1)
p(A ≥ j + 1)

proving the “⇐” direction as well. �

References

Boyd, R. N. (1983). On the Current Status of the Issue of Scientific Realism. Erkenntnis,
19:45–90.

Chakravartty, A. (2011). Scientific Realism. In Stanford Encyclopedia of Philosophy.

Crupi, V. (2013). Confirmation. The Stanford Encyclopedia of Philosophy.

Dawid, R. and Hartmann, S. (2015). The No Miracles Argument Without Base Rate

Fallacy.

Dawid, R., Hartmann, S., and Sprenger, J. (2015). The No Alternatives Argument. The
British Journal for the Philosophy of Science, 66:213–234.

Fahrbach, L. (2009). The pessismistic meta-induction and the exponential growth of.

In Hieke, A. and Leitgeb, H., editors, Reduction and elimination in philosophy and the
sciences. Proceedings of the 31th international Wittgenstein symposium.

Fahrbach, L. (2011). How the growth of science ends theory change. Synthese, 108:139–
155.

Fitelson, B. (2001). Studies in Bayesian Confirmation Theory. PhD thesis, University of
Wisconsin - Madison.

Forster, M. R. and Sober, E. (1994). How to Tell when Simpler, More Unified, or Less

Ad Hoc Theories will Provide More Accurate Predictions. British Journal for the
Philosophy of Science, 45:1–35.

20



Frigg, R. and Hartmann, S. (2012). Models in science. In Stanford Encyclopedia of
Philosophy.

Good, I. (2009). Good Thinking. Dover, Mineola, NY.

Goodman, S. N. (1999). Toward evidence-based medical statistics. 1: The P value

fallacy. Annals of internal medicine, 130:995–1004.

Hacking, I. (1965). Logic of statistical inference. Cambridge University Press, Cambridge.

Hartmann, S. and Sprenger, J. (2010). Bayesian Epistemology. In Pritchard, D., editor,

Routledge Companion to Epistemology, pages 609–620. Routledge, London.

Howson, C. (2000). Hume’s Problem: Induction and the Justification of Belief. Oxford
University Press, Oxford.

Howson, C. (2013). Exhuming the No-Miracles Argument. Analysis, 73:205–211.

Kuhn, T. S. (1977). The Essential Tension: Selected Studies in Scientific Tradition and Change.
Chicago University Press, Chicago.

Laudan, L. (1981). A confutation of convergent realism. Philosophy of Science, 48:19—-
48.

Leitgeb, H. (2014). The stability theory of belief. Philosophical Review, 123:131—-171.

Lipton, P. (2004). Inference to the Best Explanation. Routledge, New York, NY, 2nd
edition.

Lloyd, E. A. (2014). Natural Selection. In Stanford Encyclopedia of Philosophy.

Magnus, P. and Callender, C. (2004). Realist Ennui and the Base Rate Fallacy. Philosophy
of Science, 71:320–338.

Meadows, A. (1974). Communication in science. Butterworths, London.

Monton, B. and Mohler, C. (2012). Constructive Empiricism. In Stanford Encyclopedia
of Philosophy.

Psillos, S. (1999). Scientific Realism: How Science Tracks Truth. Routledge, London.

Psillos, S. (2009). Knowing the structure of nature: Essays on realism and explanation.
Palgrave Macmillan, London.

21



Putnam, H. (1975). Mathematics, Matter, and Method, volume I of Philosophical Papers.
Cambridge University Press, Cambridge.

Royall, R. (1997). Statistical Evidence: A Likelihood Paradigm. Chapman & Hall, London.

Stanford, K. (2006). Exceeding our Grasp: Science, History, and the Problem of Unconceived
Alternatives. Oxford University Press, Oxford.

Tulodziecki, D. (2015). Realist continuity, approximate truth, and the pessimistic meta-

induction.

van Fraassen, B. C. (1980). The Scientific Image. Oxford University Press, New York,
NY.

Weisberg, M. (2007). Who is a Modeler? The British Journal for the Philosophy of Science,
58:207–233.

Worrall, J. (1989). Structural Realism: The Best of Both Worlds? Dialectica, 43:99–124.

22