1. Introduction

Consider a gas that is adiabatically isolated from its environment and confined to the left half of a container. Then remove the wall separating the two parts. The gas will immediately start spreading and soon be evenly distributed over the entire available space. The gas has approached equilibrium. Thermodynamics (TD) characterizes this process in terms of an increase of thermodynamic entropy, which attains its maximum value at equilibrium. The second law of thermodynamics captures the irreversibility of this process by positing that in an isolated system such as the gas entropy cannot decrease. The aim of statistical mechanics (SM) is to explain the behavior of the gas and, in particular, its conformity with the second law in terms of the dynamical laws governing the individual molecules of which the gas is made up. In what follows these laws are assumed to be the ones of Hamiltonian classical mechanics.

We should not, however, ask for an explanation of the second law literally construed. This law is a universal law and as such cannot be explained by a statistical theory. But this is not a problem because we can rest content if we explain the ‘Boltzmannian version’ of the second law (Callender Reference Callender1999), which I call ‘Boltzmann's Law’ (BL):

What notion of probability is invoked in BL and what reasons do we have to believe that the claim it makes is true? The orthodox answer is that probabilities are time averages and that entropy is likely to increase because, assuming that the system is ergodic, the system is in equilibrium most of the time. This view is now widely believed to be untenable due to both conceptual problems and its invocation of ergodicity (see Earman and Rédei Reference Earman and Rédei1996 and van Lith Reference van Lith2001 for discussions). A propensity interpretation of SM probabilities is ruled out by the fact that the underlying microtheory, Hamiltonian mechanics, is deterministic, which is incompatible with there being propensities (Clark Reference Clark, Bricmont, Dürr, Galavotti, Ghirardi, Petruccione and Zanghì2001). Frequentism, as von Mises himself pointed out, is problematic as an interpretation of SM probabilities because a sequence of results that is produced by the same system does not satisfy the independence requirements of a collective (van Lith Reference van Lith2001, 587). Finally, so-called no-theory theories do not improve the situation because at least in the context of physical theories they do not provide an independent alternative to other accounts (Frigg and Hoefer Reference Frigg and Hoefer2007).

In two recent papers Loewer (Reference Loewer2001, Reference Loewer2004) has suggested that the way out of this deadlock is to build on David Lewis's (Reference Lewis1986, Reference Lewis1994) approach and interpret SM probabilities as chances in Lewis's sense. In this essay I first give a precise formulation of Loewer's proposal, then raise two fundamental objections, and finally conclude that these can be overcome only at the price of interpreting SM probabilities epistemically.

2. Boltzmannian Statistical Mechanics

The microstate of a system consisting of $n$ particles is specified by a point in its 6n-dimensional phase space $\Gamma $, which is endowed with the Lebesgue measure $\mu _{L}$ (its ‘natural’ measure).Footnote ¹ The dynamics of the system is governed by Hamilton's equations of motion, which define a measure-preserving phase flow $\phi _{t}$ on $\Gamma $; that is, $\phi _{t}$: $\Gamma \rightarrow \Gamma $ is a one-to-one mapping for every real number t and $\mu _{L}( \phi _{t}( B) ) =\mu _{L}( B) $ for every measurable set $B\subseteq \Gamma $. In what follows we assume that the relevant physical process begins at a particular instant $t_{0}$, and I adopt the convention that ‘$\phi _{t}( x) $’ denotes the state of the system at time $t_{0}+t$ if it was in state x at $t_{0}$, and likewise for ‘$\phi _{t}( B) $’. Similarly, ‘$\phi _{-t}( x) $’ denotes the state at time $t_{0}$ that gets mapped onto x at time $t_{0}+t$ under the dynamics of the system, and likewise for $\phi _{-t}( B) $. In a Hamiltonian system energy is conserved and hence the motion of the system is confined to the $6n-1$ dimensional energy hypersurface $\Gamma _{E}$. The measure $\mu _{L}$ can be restricted to $\Gamma _{E}$, which induces a natural and invariant measure $\mu $ on $\Gamma _{E}$ (Kac Reference Kac1959,63).

The macrostates $M_{k}$, $k=1,\ldots ,\, m$, of the system (where $m< \infty $), characterized by the values of macroscopic parameters, are assumed to supervene on the system's microstates. Therefore, each $M_{k}$ is associated with a region $\Gamma _{M_{k}}\subseteq \Gamma _{E}$ so that the system is in macrostate $M_{k}$ at $t$ if and only if its microstate x at t lies within $\Gamma _{M_{k}}$. The $\Gamma _{M_{k}}$ form a partition of $\Gamma _{E}$, meaning that they do not overlap and jointly cover $\Gamma _{E}$.

By definition, $S_{B}( M_{k}) \mathrel{\mathop:}= k_{B}\mathrm{log}\,[ \mu ( \Gamma _{M_{k}}) ] $ is the Boltzmann entropy of macrostate $M_{k}$ (where $k_{B}$ is the Boltzmann constant). Because the $\Gamma _{M_{k}}$ do not overlap, it follows that a system is in exactly one macrostate at any given time t, and for this reason it makes sense to talk about the Boltzmann entropy $S_{B}( t) $ of a system at time t: $S_{B}( t) \mathrel{\mathop:}= k_{B}\mathrm{log}\,[ \mu ( \Gamma _{M_{t}}) ] $, where $M_{t}$ is the system's macrostate at time t (i.e., $M_{t}$ is the $M_{k}$ for which it is the case that $x\in \Gamma _{M_{k}}$, where x is the system's microstate at t). The Boltzmann entropy assumes its maximum for the equilibrium state.

To rationalize Boltzmann's law we need to introduce probabilities. The standard way to do this is by appeal to the so-called Statistical Postulate (SP):

Now consider the set F of all microstates in $\Gamma _{M_{t}}$, which in the near future evolve toward macrostates $M^{\prime }$, which have higher Boltzmann entropy than $M_{t}$. With the assumption that $\mu ( F) / \mu ( \Gamma _{M_{k}}) \approx 1$ for all k except for the equilibrium state, it follows from SP that for all t the system is highly likely to evolve toward a state of higher entropy, which is exactly what BL asserts.

Whether or not this assumption holds true in a particular system is a substantial question. However, even if it does, there is a problem. It follows from the time reversal invariance of Hamilton's equations of motion that if it is true that the system is overwhelmingly likely to evolve toward a macrostate of higher entropy in the future, it is also overwhelmingly likely to have evolved into the current macrostate from a past macrostate $M^{\prime \prime }$ that also has higher entropy. This flies in the face everyday experience and leads to wrong retrodictions.

Albert (Reference Albert2000, 71–96) suggests fixing this problem by first taking the system under investigation to be the entire universe and then adopting the so-called Past Hypothesis (PH), the postulate that the universe came into being in a low entropy macrostate, the Past State, which is provided to us by modern Big Bang cosmology. The problems with flawed retrodictions can then be avoided by conditionalizing on the Past State. From a technical point of view, this amounts to replacing SP with what I call the ‘Past Hypothesis Statistical Postulate’ (PHSP):

In what follows I refer to these probabilities as ‘PHSP probabilities’. This principle is used to make predictions about the system's future by choosing B to be the set of those microstates that behave in the desired way. For instance, if we choose B to be the set F (as defined above), $p_{t}( F) $ is the probability that the system's entropy will increase in the near future given PH. If this probability comes out high for all $M_{k}$ again, except for the equilibrium state, we have explained BL. Again, whether or not this is the case is a substantive question having to do with both the construction of the macrostates as well as the dynamics of the system. However, the problems I discuss in this essay are orthogonal to this issue, and so I assume for the sake of argument that this assumption bears out in systems of interest.Footnote ²

3. L-Chances

The basis for Lewis's theory of probability is the so-called Humean mosaic, the collection of all nonmodal and nonprobabilistic actual events making up the world's entire history (from the very beginning to the very end) and upon which all other facts supervene. Lewis himself suggested that the mosaic consists of space-time points plus local field quantities representing material stuff. In a classical mechanical system the Humean mosaic simply consists of the trajectory of the system's microstate in phase space, on which the system's macrostates supervene.

The next element of Lewis's theory is a thought experiment. To make this explicit—more explicit than it is in Lewis's own presentation—I introduce a fictitious creature, Lewis's Demon. In contrast to human beings who can only know a small part of the Humean mosaic, Lewis's demon knows the entire mosaic. The demon now formulates various deductive systems that make true assertions about what is the case and, perhaps, also about what the probability for certain events are. Then the demon is asked to choose the best among these systems. The laws of nature are the true theorems of this system, and the chances for certain events to occur are what the probabilistic laws of the best system say they are (Lewis Reference Lewis1994, 480). Following Loewer, I call probabilities thus defined L-chances.

The best system is the one that strikes the best balance between strength, simplicity, and fit. The notions of strength and simplicity are given to the demon and are taken for granted in this context, but the notion of fit needs explicit definition. Every system assigns probabilities to certain courses of history, among them the actual course; the fit of the system is measured by the probability that it assigns to the actual course of history, that is, by how likely it regards things that actually happen. By definition systems that do not involve probabilistic laws have perfect fit. As an illustration, consider a Humean mosaic that consists of just 10 outcomes of a coin flip: HHTHTTHHTT. Theory $T_{1}$ posits that all events are independent and sets $p( H) =p( T) =0\mathrm{.}\,5$; theory $T_{2}$ shares the independence assumption but posits $p( H) =0\mathrm{.}\,9$ and $p( T) =0\mathrm{.}\,1$. It follows that $T_{1}$ has better fit than $T_{2}$ because $( 0\mathrm{.}\,5) ^{10}> ( 0\mathrm{.}\,1) ^{5}( 0\mathrm{.}\,9) ^{5}$.

Loewer's suggestion is that Boltzmannian SM as introduced above—the package of Hamiltonian mechanics, PH and PHSP—is a putative best system of the sort just described (Reference Loewer2001, 618; Reference Loewer2004, 1124) and that PHSP probabilities can therefore be regarded as L-chances. But there is an obvious problem, namely, reconciling determinism and the existence of probabilistic laws, which Lewis himself thought was impossible (Reference Lewis1986, 118).

Loewer claims that Lewis was wrong about this and suggests that introducing probabilities via initial conditions solves the problem:

Loewer does not tell us what exactly he means by “a kind of dynamical chance,” in what sense this chance is macroscopic, how its values are calculated, and how it connects to the technical apparatus of SM. I will now present how I think this proposal is best understood and show that, on this reading, Loewer's “macroscopic chances” coincide with PHSP as formulated above.

As in Section 2, I take the system's state at $t> t_{0}$ to be the macrostate $M_{t}$. We now need to determine the probability of the event ‘being in set $B\subseteq \Gamma _{M_{t}}$ at time t’. As I understand it, Loewer's proposal falls into two parts. The first is that the probability of an event at a time t is ‘completely determined’ by the probability of the corresponding event at time $t_{0}$; that is, the probability of the event ‘being in set B at time t’, $p_{t}( B) $, is equal to the probability of ‘being in set $B_{0}$ at time $t_{0}$’ where $B_{0}$ is, by definition, the set that evolves into B under the dynamics of the system after time t has elapsed. Formally, $p_{t}( B) =\mu _{0}( B_{0}) =\mu _{0}( \phi _{-t}( B) ) $, where $\mu _{0}$ is the microcanonical distribution over the Past State, that is, $\mu _{0}( \:\cdot \:) =\mu ( \:\cdot \:\cap \Gamma _{M_{p}}) / \mu ( \Gamma _{M_{p}}) $.

The second part is conditionalizing on the entire macrohistory up to time t, that is,a specification of the system's macrostate at each instant of time between $t_{0}$ and t. A possible macrohistory, for instance, is that system is in macrostate $M_{1}$ during the interval $[ t_{0},\; t_{1}] $, in $M_{5}$ during $( t_{1},\: t_{2}] $, in $M_{7}$ during $( t_{2},\: t_{3}] $, and so forth, where $t_{1},\, t_{2},\, t_{3},\,\ldots $ are the instants of time at which the system changes from one macrostate into another. What we are now expected to calculate is the probability of ‘being in set B at time t’ given the system's macrohistory. Let $Q_{t}$ be the set of all microstates in $\Gamma _{M_{t}}$ that are compatible with the entire past history of the system; that is, it is the set of all $x\in \Gamma _{M_{t}}$ that lie on trajectories that for every t were in the $\Gamma _{M_{k}}$ corresponding to the actual macrostate of the system at t. The sought after conditional probability then is $p_{t}( B\vert Q_{t}) =p_{t}( B\& Q_{t}) / p_{t}( Q_{t}) $, provided that $p_{t}( Q_{t}) \neq 0$, which, as we shall see, is the problematic condition.

Putting these two parts together we obtain the fundamental equation defining L-chances for deterministic systems:

where, again, $\mu _{0}( \:\cdot \:) =\mu ( \:\cdot \:\,\cap \,\Gamma _{M_{p}}) / \mu ( \Gamma _{M_{p}}) $.

The crucial thing to realize now is that, due to the conservation of the measure, the expression for the conditional probability in PHSP can be expressed as $p_{t}( B\vert R_{t}) =\mu ( \phi _{-t}( B\cap R_{t}) ) / \mu ( \phi _{-t}( R_{t}) ) $. Trivially, we can substitute $\mu _{0}$ for $\mu $ in this expression, which makes it equivalent to Equation (1) if we treat $Q_{t}$ and $R_{t}$ as equals. (In fact, there is a difference between them in that $R_{t}$ only involves a conditionalization on PH, while $Q_{t}$ contains the entire past history. However, nothing in PHSP depends on this, and one could just as well include the entire history in $R_{t}$.) Hence, PHSP can be interpreted as attributing probabilities to events at $t> t_{0}$ solely on the basis of the microcanonical measure over the initial conditions, which is precisely what Loewer needs.

4. Problems with Fit

Loewer claims that SM as introduced above is the system that strikes the best balance between simplicity, strength, and fit. Trivially, this implies that it can be ranked along these three dimensions. Simplicity and strength are no more problematic in SM than they are in any other context, and I shall therefore not discuss them further here. The problematic concept is fit.

The fit of a theory is measured in terms of the probability that it assigns to the actual course of history. But what history? Given that L-chances are calculated using the Lebesgue measure, which assigns measure zero to any trajectory, they do not lead to a nontrivial ranking of microhistories (trajectories in $\Gamma $). The right choice seems to be to judge the fit of theory with respect to the system's macrohistory.

What is the probability of a macrohistory? A first answer to this question would be to simply use Equation (1) to calculate the probability of a macrostate at each instant of time and then multiply them all, just as we did in the above example with the coins (with the only difference that the probabilities are now not independent any more, which is accounted for in Equation [1]). This is plain nonsense. There is an uncountable infinity of such probabilities and multiplying an uncountable infinity of numbers is an ill-defined operation. Determining the probability of a history by multiplying probabilities for individual events in the history works fine as long as the events are discrete (like coin flips), but it fails when we have a continuum.

Maybe this was too crude a stab at the problem, and when taking the right sorts of limits things work out fine. Let us discretize time by dividing the real axis into small intervals of length $\delta $, then calculating the probabilities at the instants $t_{0}$, $t_{0}+\delta $, $t_{0}+2\delta $, and so forth, multiply them (there are only countably many now), and then take the limit $\delta \rightarrow 0$. This would work if the $p_{t}( B\vert Q_{t}) $ depended in a way on $\delta $ that would assure that the limit exists. This is not the case. In fact, for all $t> t_{1}$ (i.e., after the first change of macrostate), the $p_{t}( B\vert Q_{t}) $ do not exist because $Q_{t}$ has measure zero, and this irrespective of $\delta $. This can be seen as follows. Take the above example of a macrohistory and consider an instant $t\in ( t_{1},\: t_{2}] $ when the system is in macrostate $M_{5}$. To calculate the probability of the system being in $M_{5}$ at t we need to determine $Q_{t}$, the set of all microstates in $\Gamma _{M_{5}}$ compatible with the past macrohistory. Now, these points must be such that they were in $M_{1}$ at $t_{1}$ and in $M_{5}$ just an instant later (i.e., for any $\varepsilon > 0$, at $t_{1}+\varepsilon $ the system's state is in $\Gamma _{M_{5}}$). The mechanical systems we are considering have global solutions (or at least solutions for the entire interval $[ t_{0},\: t_{f}] $, where $t_{f}$ is the time when the system ceases to exist), and trajectories in such systems have finite phase velocity; that is, a phase point x in $\Gamma $ cannot cross a finite distance in no time. From this it follows that the only points that satisfy the condition of being in $M_{1}$ at $t_{1}$ and in $M_{5}$ just instant later are the ones that at $t_{1}$ lie exactly on the boundary between $M_{1}$ and $M_{5}$. But the boundary of a $6n-1$ dimensional region is $6n-2$ dimensional and therefore has measure zero. Therefore $Q_{t}$ has measure zero for all $t> t_{1}$, and accordingly $p_{t}( B\vert Q_{t}) $ does not exist for $t> t_{1}$, no matter what B is. Needless to say, this renders the limit $\delta \rightarrow 0$ obsolete.

The source of the problem is the conjunction of three elements: (1) the posit that time is continuous, (2) the assumption that the transition from one macrostate to another one takes place at a precise instant, (3) the posit that we conditionalize on the entire macrohistory of the system. We have to give up at least one of these to obtain nonzero $p_{t}( B\vert Q_{t}) $. The problem is that all three elements either seem reasonable or are deeply entrenched in the theory and cannot be renounced without far-reaching consequences.

The first option, discretizing time, would solve the problem because, if we assume that time is discrete, the macrohistory is discrete too. If we only consider, say, the events ‘being in $\Gamma _{M_{1}}$ at instant $\tau _{1}$’ and ‘being in $\Gamma _{M_{5}}$ at instant $\tau _{2}$’, where $\tau _{1}\in [ t_{0},\: t_{1}] $ and $\tau _{2}\in ( t_{1},\: t_{2}] $, sets of finite measure can move from $\Gamma _{M_{1}}$ to $\Gamma _{M_{5}}$, and $Q_{t}$ no longer needs to have measure zero. The problem with this suggestion is that it is ad hoc and defeats the purpose of SM. If we believe that classical mechanics is the fundamental theory governing the microconstituents of the universe and set out to explain the behavior of the universe in terms of its laws, not much seems to be gained if such an explanation can only be had at the expense of profoundly modifying these laws.

The second suggestion would be to allow for finite transition times between macrostates, that is, allowing for there to be periods during which it is indeterminate in which macrostate the system is. This suggestion is not without merit, as one could argue that sharp boundaries between macrostates are indeed a mathematical idealization that is ultimately unjustifiable from a physics perspective. However, sharp transitions are a direct consequence of the postulate that macrostates supervene on microstates. This postulate is central to the Boltzmannian approach, and it is not clear how it could be given up without unsettling the entire edifice of Boltzmannian SM.

The third option denies that we should conditionalize on the complete macrohistory. The idea is that even though time at bottom is continuous, the macrohistory takes record of the system's macrostate only at discrete instants and is oblivious about what happens between these. That is, what we should conditionalize on is the discrete macrohistory (DMH): $M_{p}$ at $\tau _{0}$, $M_{\tau _{1}}$ at $\tau _{1}$, $M_{\tau _{2}}$ at $\tau _{2}\ldots ,$ $M_{\tau _{j-1}}$ at $\tau _{j-1}$, and $M_{\tau _{j}}$ at $\tau _{j}$, where $t_{0}=\mathrel{\mathop:} \tau _{0}\leq \tau _{1}\leq \ldots \leq \tau _{j-1}\leq \tau _{j}\mathrel{\mathop:}= t_{f}$ is a finite number of instants of time and the $M_{\tau _{i}}$ the system's macrostate at time $\tau _{i}$.Footnote ⁴ This solves the problem because, when conditionalizing on a discrete macrohistory, $Q_{t}$ no longer necessarily is of measure zero.

This is at once the most feasible and the most problematic suggestion. It is feasible because it does not require revisions in the structure of the theory. It is problematic because we have given up the notion that the fit of a theory has to be best with respect to the complete history of the world and replaced it with the weaker requirement that fit be best for a partial history. (And mind you, the point is not that the fit of the full history is in practice too complicated to calculate and we therefore settle for a more tractable notion; the point is that the fit of a complete macrohistory is simply not defined because the relevant conditional probabilities do not exist.) From the point of view of Lewis's theory this seems unmotivated. Fit, like truth, is a semantic concept characterizing the relation between the theory and the world, and if the Humean mosaic has continuous events in it, there should be a matter of fact about what the fit of the theory is.

Moreover, even if one was willing to believe that a discrete version of fit was satisfactory, it is not clear whether this leads to useful results. Depending on which particular instants of time one chooses to measure fit, one can get widely different results. Conditionalizing on a DMH would be useful only if it was the case that the fit rankings came out the same no matter what choice of instants we make. There is at least a question whether this is the case.

5. The Putative Best System Is Not the Best System

I now assume that a DMH notion of fit can be defended in one way or another.Footnote ⁵ Then a further problem emerges: the package consisting of Hamiltonian Mechanics, PH, and PHSP in fact is not the best system. The reason for this is that we can always improve the fit of a system if we choose a measure that, rather than being uniform over $\Gamma _{M_{p}}$, is somehow peaked over those initial conditions that are compatible with the entire DMH.

Let us make this more precise. The probability of the DMH is $p( DMH) =p_{\tau _{0}}( B_{\tau _{0}}\vert Q_{\tau _{0}}) \,\ldots \, p_{\tau _{j-1}}( B_{\tau _{j-1}}\vert Q_{\tau _{j-1}}) $, where the $B_{\tau _{i}}$ are those subsets of $\Gamma _{M_{\tau _{i}}}$ that evolve into $\Gamma _{M_{\tau _{i+1}}}$ under the evolution of the system. One can then prove that

where $\Gamma _{p}\mathrel{\mathop:}= \Gamma _{M_{p}}$ and $\Gamma _{i}\mathrel{\mathop:}= \Gamma _{M_{\tau _{i}}}$ for $i=1,\ldots ,\,\, j$. Now define $N\mathrel{\mathop:}= \Gamma _{p}\cap \phi _{-\tau _{1}}( \Gamma _{1}) \,\ldots \,\phi _{-\tau _{j}}( \Gamma _{j}) $. The fit of system is measured by the probability that it assigns to the actual DMH, which is given by Equation (2). It is a straightforward consequence of this equation that the fit of a system can be improved by replacing $\mu _{0}$ by a measure $\mu _{P}$ that is peaked over $N$. Fit becomes maximal (i.e., $p( DMH) =1$) if, for instance, we choose the measure $\mu _{N}$ that assigns all the weight to N and none to $\Gamma _{p}\setminus N$. Trivially, $N$ contains the actual initial condition of the universe. A simpler and more convenient distributions that yields maximal fit is a Dirac delta function over the actual initial condition.

If there is such a simple way to improve (and even maximize) fit, why does the demon not provide us with a system comprising $\mu _{N}$ or a delta function? Coming up with such a system is not a problem for the demon, as, by assumption, the demon knows the entire Humean mosaic, which contains the exact initial condition.

A reason to prefer $\mu _{0}$ to other measures might be that these make the system less simple and that this loss in simplicity is not compensated by a corresponding gain in fit and strength. This seems implausible. Handling a Dirac delta function rather than $\mu _{0}$ does not render the system more complicated while the gain in fit is considerable. Hence, simplicity does not seem to provide reason to prefer $\mu _{0}$ to other measures that have better fit.

6. Outlook: Epistemic Probabilities after All

The system consisting of Hamiltonian mechanics, PH, and PHSP is not the best system, and therefore PHSP probabilities cannot be interpreted as L-chances. In this section I first want suggest that the probabilities in this system are best understood as epistemic probabilities of sorts and then indicate how this view could be defended against some common objections.

Every theory involving probabilities must answer the question of what these probabilities are probabilities for. The initial conditions approach to chance does not seem to have an answer to this question. The universe has exactly one initial microcondition, and there is nothing chancy about this condition. How, then, can we understand a probability distribution over initial conditions? The only answer seems to be that this distribution reflects our ignorance about the systems's initial microcondition; all we know is the system's initial macrostate, and so we put a probability distribution over the microconditions compatible with that macrostate that reflects our lack of knowledge.

How these epistemic probabilities should be understood is a question that I cannot discuss here. Let me just indicate that there are at least two options. The first, a version of objective Bayesianism, appeals to Jayenes's maximum entropy principle, which indeed instructs us to prefer $\mu _{0}$ to alternative measures because, given the information about the system's macrostate, $\mu _{0}$ maximizes the (continuous) Shannon entropy. The other alternative is to revise Lewis's account in a way that builds epistemic restrictions of the users of theories into the selection criteria for systems. Hoefer's (Reference Hoefer2007) theory of Humean chance makes room for this possibility.

There are two main complaints about an epistemic interpretation of SM probabilities. The first points out that the thermodynamic entropy is a property of a physical system and that $S_{B}$ coincides with it up to a constant. This, so the argument goes, is inexplicable on the basis of an epistemic approach to probabilities.Footnote ⁶ This is wrong because $S_{B}$ is defined in terms of the measure of certain chunks of phase space, and probabilities (no matter how we interpret them) have simply nothing to with it.

The second complaint concerns the alleged causal efficacy of human knowledge. The point becomes clear in the following—rhetorical—questions by Albert (Reference Albert2000, 64):Footnote ⁷

It can’t be, and no one should think that it could. Proponents of epistemic probabilities need not believe in parapsychology. What underlies this objection is the mistaken view that PHSP probabilities play a part in bringing about things in the world. Of course the cooling down of drinks and the boiling of kettles has nothing to do with what anybody thinks or knows about them, but they have nothing to do with the probabilities attached to these events either. Drinks cool down and kettles boil because the universe's initial condition is such that under the dynamics of the system it evolves into a state in which this happens. All we need to explain why things happen is the initial condition and the dynamics.

Last but not least, the decision to conditionalize on DMH rather than the full macrohistory seems to square better with an epistemic approach to probabilities. For these reasons I suggest that we take seriously the option to interpret SM probabilities epistemically.