I refer, of course, to bodies of explanation. There are two such bodies
named ‘statistical’ (sometimes ‘probabilistic’), differing fundamentally
in their structure, where the name suggests that there is only one. The
extra body is therefore rarely noticed.
The leading accounts that philosophers, including Peirce, have given
of statistical explanation all agree, amazingly, in ignoring the special
character of explanation in statistical mechanics (Railton, vide infra,
excepted).17 Contemporary discussions of this topic begin with the ‘covering
law’ model due to Hempel (1962), and it is convenient for us to
begin there, too.
According to Hempel, statistical explanation is an inference, of a statement
of the phenomenon to be explained, from premisses that include
a statement of at least one probabilistic law. Such a law, in the simplest
case, is of the form P(E /C ) = p, where p is a real number, 0 ≤ p ≤ 1, and
P(E /C ) is the probability of E conditional on C obtaining. Thus, from a
premiss that C and a premiss that P(E /C ) = p, we can infer that E, the
inference having a probability p of being correct, relative to the information
supplied in the premisses.
Others deny that explanation is inference, and deal far differently
from Hempel with the problem of the reference class (i.e., how C is to
17 The literature on statistical explanation should not be confused with the much larger and more sophisticated literature on statistical inference. Although the former sometimes draws on the latter, the latter is not concerned with what constitutes explanation; it is concerned with the meaning of probability and the different kinds of inference that different theories of probability justify. Those issues are central to foundational debates in statistical mechanics, but relatively little of the literature on statistical inference deals with statistical mechanics, von Mises (1957 [1928]) and Jeffreys (1973 [1931]) being major exceptions.
be chosen) and the question of how, if at all, assessing the probability
of an outcome either contributes to or is implicated in its explanation.
Richard Jeffrey (1970) pointed out that assigning an explanandum a
high probability does not always explain it and, more surprisingly, that
some good explanations accord the explanandum a low probability. If
things of a certain kind happen by chance (e.g., the decay of an atom of a
radioactive element in a given period of time), then that is how a thing of
that kind happens, even if its chance was low. But all agree that statistical
explanations presuppose probabilistic laws of the sort described – laws
that enable us to derive a probability for a given outcome from known
conditions.
And that fails utterly to capture the reasoning in statistical mechanics,
in which the laws assumed may be deterministic, not probabilistic,
and in which the initial conditions are unknown. A typical example is the
explanation of the evolution of an enclosed system of gas molecules from
being less equally to being more equally distributed. There are trillions of
molecules in even a cubic centimeter of gas, and any observable distribution
of them (a macrostate) would be constituted by any of an enormous
number of alternative arrangements (microstates) of the molecules. We
must therefore remain ignorant of the actual arrangements, including
the initial arrangement. Statistical reasoning shows nonetheless that the
chances overwhelmingly favor changes from less to more equal distribution,
until near-equality is reached. Rather than deriving a probability
from known conditions, the movement of thought in statistical mechanics
is almost the polar opposite: from ignorance of initial conditions (at the
microlevel) to virtual certainty about the outcome (at the macrolevel).18
How is it possible that this obvious point has been overlooked or,
at least, neglected? The explanation, in part, is that the formalism
P(E /C ) = p does not distinguish between the two kinds of case. Thus,
Hempel cited a range of examples, from radioactive decay to rolling
dice, without distinguishing probabilistic dependence of an outcome on
known conditions (radioactive decay) from ignorance of conditions possibly
deterministic (rolling dice) (1962, pp. 121–2).
Perhaps the best-known alternatives to Hempel’s model are two due to
the lateWesley Salmon. In his ‘statistical relevance’ model, a fact explains
an event if, putting it perhaps too simply, it accords it a probability that
would be unaltered by any further fact other than that of the event
itself (1970). Salmon is quite clear that such assessments of probability
18 See chapter 5, section 1, for a more detailed discussion of this example.
presuppose probabilistic laws (1970). No distinction is made between different
forms of statistical explanation.19 Salmon’s later, ‘causal/ mechanical’
model retains the idea of statistical relevance but adds to it the
requirement of a causal/mechanical account, that is, some idea of the
processes that lead probabilistically to the explained result (1984, p. 22).
Here, at last, it becomes possible to make the needed distinction between
mechanistic and other forms of statistical explanation, but Salmon did
not make it. Instead, it is James Woodward who pointed out that the
causal/mechanical model is not satisfied in statistical mechanics. For
in that science, Woodward says, ‘one abstracts radically from details
of such individual causal processes and focuses on finding a way of
representing the aggregate behavior of molecules’ (Woodward 1989,
p. 363).
Peter Railton’s is the only model of statistical explanation – or probabilistic
explanation, in his preferred term – that I know of, that explicitly
excludes statistical mechanics and thereby makes the needed distinction
(Railton 1977). He calls the explanations he models ‘deductivenomological-
probabilistic’ (D-N-P) explanations. We can skip over the
reason Railton gives why they are deductive. The thesis germane to our
interest is that D-N-P explanations must be causal (in Railton’s sense of
‘causal’, i.e., they must cite mechanisms) as well as probabilistic: they are
‘unsatisfactory unless we can back them up with an account of the mechanism(
s) at work’ (p. 208). Such a mechanism, not being deterministic,
is a ‘chance mechanism’, and thus the model is restricted to genuinely
indeterministic processes:
It is widely believed that the probabilities associated with standard gambling
devices, classical thermodynamics, actuarial tables, weather forecasting, etc., arise
not from any underlying physical indeterminism, but from an unknown or uncontrolled scatter of initial conditions. If this is right, then D-N-P explanation would
be inapplicable to these phenomena even though they are among the most familiar
objects of probabilistic explanation. I do not, however, find this troublesome:
if something does not happen by chance, it cannot be explained by chance.
(p. 223)
19 It is remarkable that Salmon, like Hempel and many other authors in this field, paid so little attention, at least in this context, to the explanations yielded by statistical mechanics. Salmon cited statistical mechanics several times, but always briefly and with respect to examples in which thermodynamic laws interpreted probabilistically are assumed, not explained (1970, pp. 209ff.; 1984, pp. 26, 180–1; 1998, p. 151). But the chief glory of statistical mechanics is its explanation of those laws: a fact of which these same authors were well aware and that has been much discussed in another context by philosophers of science, where the topic is theoretical ‘reduction’.
It might be objected that explaining something as being due to chance is
no explanation at all; but Railton’s thought is that it is the chance mechanism
that explains its effects, probabilistically, and not chance per se.20
Railton continues, ‘What must be given up is the idea that explanations
can be based on probabilities that have no role in bringing the world’s
explananda about’ (p. 223, emphasis in original).21 But that is an amazing
claim. For, surely, statistical mechanics, even in its early, Newtonian
phase, has provided some of the most impressive and successful explanations
in modern science. They have been called ‘explanations’ and have
been accepted as such and felt to be explanatory. To deny that they are
explanations is, in effect, to impose one’s narrower definition in lieu of
a broader, established use of the term. Far better, I think, to admit that
explanation takes different forms.
Explanations we may call ‘statistical’ fall into two classes. Those of
the one class are the explanations that Hempel, Salmon, Railton, et al.
evidently had in mind when they offered their models of statistical or
probabilistic explanation. They are mechanistic in kind; for they all seek
to explain particular outcomes by citing particular conditions. By our
generous definition of ‘mechanistic’, they are so even if they lack the
specification of mechanisms that Railton and Salmon say is essential to
a complete mechanistic explanation. Let us follow Railton, but without
insisting that mechanisms be specified, in naming this subclass of statistical
explanation ‘probabilistic explanation’.
The other class of statistical explanation is not mechanistic; at least,
that is what I argue in some detail in the next chapter with respect, first, to
statistical mechanics and, next, to natural selection. That of course contradicts
the standard view that both sciences are mechanistic. Neglected
by logicians, this latter form of statistical explanation needs a name. As
the explananda of these explanations are anisotropic processes, that is,
the (practically) irreversible evolution of systems toward final states or
toward new states, let us call them ‘anisotropic explanations’.
20 This is in line with the growing tendency of philosophers to acknowledge a probabilistic
causality (Suppes 1970; Fetzer and Nute 1979), in contrast to the past fashion, of claiming
that quantum mechanics spelled causality’s demise. I believe that Peirce would have
welcomed this development, as witness his later theory of probability as a disposition
(8.225, 2.664–5; cf. Burks 1964), which anticipated Popper’s well-known propensity
theory (Popper 1959) that Railton exploits. Deterministic causality may then be seen as
a special case of probabilistic causality (where p=1).
21 In a later paper, Railton acknowledges that classical, i.e., Newtonian, statistical mechanics is explanatory, but only as it bears on an ideal but unavailable explanation that, in this case, would be deterministic (1981, pp. 249–52).
The division of explanations between those that are deterministic
(‘deductive nomological’, in Hempel’s phrase) and those that are statistical
is far less fundamental, I suggest, than is the distinction between
mechanistic explanations, whether deterministic or probabilistic, on the
one hand, and anisotropic explanations, on the other. The use of statistics
is so different in probabilistic and anisotropic explanations, respectively,
that the two being lumped together as ‘statistical’ is misleading though
not incorrect. In the next chapter, I argue that teleological explanation is
a subclass of anisotropic explanation. Teleological explanation is distinct
from explanation in statistical mechanics, yet the two are far closer in
nature to one another than either is to probabilistic explanation.
The failure to recognize that explanation in statistical mechanics is not
probabilistic (in the sense we are now giving to the latter term), but is of
another form altogether, has kept that form of explanation from being
recognized. And, as teleological explanation is, at bottom, of that form,
it has kept teleological explanation under its historic cloud of suspicion.
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