Microsoft Word - Dunn - Fried Eggs _Writing Sample_


 1

Fried Eggs, Thermodynamics, and the Special Sciences 

(forthcoming in British Journal for the Philosophy of Science) 

Jeffrey Dunn 

Abstract: David Lewis ([1986b]) gives an attractive and familiar account of counterfactual 

dependence in the standard context. This account has recently been subject to a 

counterexample from Adam Elga ([2000]). In this paper, I formulate a Lewisian response to 

Elga’s counterexample. The strategy is to add an extra criterion to Lewis’s similarity metric, 

which determines the comparative similarity of worlds. This extra criterion instructs us to 

take special science laws into consideration as well as fundamental laws. I argue that the 

Second Law of Thermodynamics should be seen as a special science law, and give a brief 

account of what Lewisian special science laws should look like. If successful, this proposal 

blocks Elga’s counterexample.   

1  Introduction 
2  The Asymmetry of Miracles and Overdetermination 
3  Elga’s Counterexample 
4 Possible Solutions 
5  Structure of the Special Science Solution 
6  The Special Science Solution 
 6.1  Thermodynamics as a Special Science 
 6.2  Lewisian Special Science Laws 
7  Objections 
8  Conclusion 
 

1 Introduction 

In Counterfactuals, David Lewis ([1973]) tells us that, in general, it is a fallacy to strengthen the 

antecedent of counterfactuals. That is, inferring ((p ∧ q) > r) from (p > r) is invalid. But 

when we evaluate counterfactuals in everyday life, we often must do something very much 

like strengthening the antecedent. If I hadn’t set my alarm last night, then what would have 



 2

happened? Without some way to strengthen the antecedent—to indicate that we don’t stray 

too far from actuality—it’s far too indeterminate to say.  

Lewis’s well-known account of how to fill in the missing information in counterfactuals 

relies on the comparative similarity between worlds. The counterfactual (p > q) is non-

vacuously true just in case in the set of all the worlds, W, where p is true there is at least one 

world, w ∈ W where q is true, and there is no world w′ ∈ W that is more or equally similar to 

the actual world than w and where q is false. Thus, the state of the most similar world at 

which the antecedent is true does the job of filling in the details left unsaid in the antecedent 

(Lewis [1986b], p. 41). 

Similarity is highly context dependent. In certain contexts one feature of a world 

matters for similarity, in another context, a different sort of feature of a world matters for 

similarity. The context dependence of similarity is a strong point of Lewis’s account: it 

mirrors the context dependence of counterfactuals. However, in ‘Counterfactual 

Dependence and Time’s Arrow’ Lewis ([1986b]) goes further with his account. Although our 

evaluation of counterfactuals can vary with context, Lewis notes that there is a standard 

context in which counterfactuals are typically evaluated. Within this context there is an 

asymmetry of counterfactual dependence: roughly, if the past had been different, the future 

would be different, but if the future were different, that wouldn’t change the past. This 

asymmetry involves two claims; I will be concerned solely with the first in this paper.  

Lewis then proposes his familiar metric that determines the similarity of worlds in 

the standard context. This is to deliver correct truth-conditions for counterfactuals and thus 

give us the asymmetry of counterfactual dependence: 

 (1) It is of the first importance to avoid big, widespread, diverse violations of law. 

(2) It is of the second importance to maximize the spatio-temporal region  



 3

throughout which perfect match of particular fact prevails. 

(3) It is of the third importance to avoid small, localized, simple violations of law. 

(4) It is of little or no importance to secure approximate similarity of particular fact. 

(Lewis [1986b], pp. 47-8)1 

Lewis’s most general insight concerning counterfactuals—that their truth-conditions 

depend on a comparative similarity relation—is very plausible, and perhaps correct. 

However, his more specific claims about similarity in the standard context and 

counterfactual dependence have come under more serious attack. In this paper I will be 

primarily concerned with one important counterexample to this similarity metric given by 

Adam Elga ([2000]) and offer what I think is a Lewisian solution. Lewis’s similarity metric 

appeals only to fundamental laws. The solution I will advocate is to tweak the metric so that 

matching of non-fundamental, special science laws is taken account of when judging the 

similarity of worlds. I will argue that this solution blocks Elga’s counterexample, and allows 

us to have the asymmetry of counterfactual dependence that Lewis desired.  

It is worth noting that this project is of wider interest than the analysis of 

counterfactuals. For a critical part of my proposed solution is to appeal to special science 

laws. Lewis, however, never offers an account of such laws, so part of my proposal will be to 

sketch a Lewisian account of such laws.2 Though I will only be focused on showing how 

such an account can be used to answer Elga’s counterexample, and thus get the asymmetry 

of counterfactual dependence, there are other profitable uses to which an account of special 

science laws could be put.  

                                                 
1 I will refer to these four constraints as the similarity metric. It is worth noting that securing a way to evaluate 
counterfactuals in the standard context is important to other areas of Lewis’s philosophy, most notably his 
analysis of causation.  
2 In “New Work For A Theory of Universals” ([1999], pp. 42-43), Lewis makes a distinction between the 
fundamental laws, which are the axioms of the best system, and the derived laws that follow fairly 
straightforwardly from the fundamental laws. This, however, does not get us the special science laws: laws that 
may have exceptions and that are not obviously entailed by the fundamental laws. 



 4

For one, Lewis’s account of laws gives us the fundamental laws: the axioms of the 

simplest, strongest systematization of the phenomena described in a language that references 

only perfectly natural properties. But there are almost certainly lawlike relations that are not 

entailed by the fundamental laws. So in looking only at fundamental laws and their 

entailments, we are apt to miss out on interesting structural facts about the world: that 

certain collections of fundamental entities move about in generally lawlike ways, ways which 

are not entailed by fundamental law. By introducing the notion of a special science laws, we 

are in a better position to track these interesting structural facts about the world.  

Another reason for wanting a Lewisian account of special science laws is because 

they will be helpful in evaluating a certain special class of counterfactuals.3 Consider the kind 

of counterfactual that a biologist might utter:  

If it were that A, but the biological laws were the same, then C would have been the case. 

What we want to do is to evaluate a certain claim in all of the A-worlds where the biological 

laws are the same. It is often irrelevant to this kind of counterfactual whether or not we have 

match of fundamental law. It is a substantive and unsupported hypothesis that any world 

with matching biological laws is one with matching fundamental laws. What is important is 

match of biological law. But then we need an account of such laws. An account of special 

science laws will be of particular service when A is, or implies, the denial of a fundamental 

law. For instance, Marc Lange ([2000a]) considers the ‘area law’ of island biogeography, 

which holds that the equilibrium number of species on an island is proportional to the area 

of the island (S = cAz).   

                                                 
3 Later in this paper I will propose a new similarity metric for the evaluation of counterfactuals in the standard 
context. It is important to note that I am not claiming that that proposed similarity metric (for the standard 
context) is capable of handling the kinds of counterfactuals discussed in this paragraph. It is plausible that 
counterfactuals like this, which explicitly mention laws, may take us out of the standard context. My claim is 
that having an account of special science laws will be useful in whatever account gets these kinds of 
counterfactuals right. 



 5

For instance, suppose there had been birds with antigravity organs that assist them a 

little in becoming airborne. […] had there existed such creatures, the island-

biogeographical laws would still have held. After all, the factors affecting species 

dispersal would have been no different. For instance, smaller islands would still have 

presented smaller targets to off-course birds or driftwood-borne seeds and so would 

have picked up fewer stray creatures as migrants. ([2000], pp. 236-7)  

We’d like to be able to deal with situations like this, where the antecedent of a counterfactual 

implies that the fundamental laws are violated. Though I will not address such counterlegals 

in this paper, an account of special science laws would be of clear relevance to such a 

project. What we’d like to have is some way of meaningfully comparing worlds, all of which 

exhibit massive violations of fundamental laws. An attractive way of dealing with such 

scenarios is to appeal to special science laws. But again, to appeal to such laws, we need an 

account of such laws.  

So, though my focus will be on Elga’s counterexample and the asymmetry of 

counterfactual dependence, I think there is independent motivation to pursue an account of 

Lewisian special science laws. I will sketch such an account, and then attempt to show its 

usefulness in responding to the problem Elga raises for Lewis’s account of the truth-

conditions of counterfactuals in the standard context. Before we get to Elga’s 

counterexample, however, there is some groundwork to be done. 

 

2 The Asymmetry of Miracles and Overdetermination 

To see Lewis’s original similarity metric in action consider the counterfactual: 

(Nuclear) If Nixon had pressed the button, there would have been nuclear holocaust. 



 6

For (Nuclear) to be true on Lewis’s account, the similarity metric must ensure that there is 

some button-pressing-nuclear-holocaust world more similar to the actual world than any 

button-pressing-no-holocaust world. According to Lewis, there is such a world: it perfectly 

matches the actual world up until just before Nixon’s (counterpart’s)4 decision to press the 

button. Assuming the actual world to be deterministic,5 and since Nixon is to press the 

button in this possible world, the laws of the actual world cannot hold in this possible world. 

There must, then, be a miracle in this possible world to bring about Nixon’s pushing of the 

button. Miracles are relativized to world-pairs: some possible world exhibits a miracle 

relative to the actual world just in case there is an event in that possible world that violates 

the laws of the actual world. The possible world under consideration has such a miracle 

relative to the actual world. After this miracle-induced button-pressing, however, the 

deterministic laws of the actual world hold, ensuring a nuclear holocaust in that possible 

world. Call this nuclear-holocaust world. Lewis’s similarity metric is designed to ensure that 

nuclear-holocaust world is more similar to the actual world than any button-pressing-no-

holocaust world. This yields asymmetric counterfactual dependence in the standard context. 

However, consider a possible world that seems to do just as well on the similarity 

metric, and yet at which the consequent of (Nuclear) is false. The past of this world is 

nothing like the past of the actual world in matter of fact, though the deterministic laws of 

the actual world are obeyed during this time. In this world, Nixon presses the button, 

uncompelled by a miracle. However, immediately after he presses the button, there is a 

                                                 
4 Given Lewis’s well-known views about possible worlds, Nixon does not literally reappear in other possible 
worlds; instead, he has counterparts in these worlds. For the rest of the paper I will drop talk of counterparts, 
but it is to be assumed that when I talk of otherworldly Nixons or otherworldly eggs (etc.), that I mean to be 
speaking of counterparts of these things. 
5 Throughout Lewis’s discussion, and subsequent criticism of Lewis’s view, it is assumed that the actual world 
is deterministic according to the Newtonian dynamical laws. Given this stipulation, we aren’t considering the 
actual world since the Newtonian laws are false here. Instead we are considering a deterministic Newtonian 
correlate of the actual world. I ignore the fact that Newtonian dynamics is not strictly deterministic (see Earman 
[1986]). 



 7

miracle that brings this world into perfect match with the actual world. Call this world 

different-past world. Thereafter, different-past world and the actual world evolve (in perfect 

match) obeying the deterministic laws of the actual world. We might wonder why different-

past world isn’t at least as similar to the actual world as nuclear-holocaust world. Both 

feature one miracle, and a long stretch of perfect match. Lewis considers worlds such as 

different-past world: 

But are there any such worlds to consider? What could they be like: how could one 

small, localized, simple miracle possibly do all that needs doing? How could it deal 

with the fatal signal, the fingerprints, the memories, the tape, the light waves and all 

the rest? I put it to you that it can’t be done! Divergence from a world such as w0 [the 

actual world] is easier than perfect convergence to it. Either takes a miracle, since w0 

is deterministic, but convergence takes much more of a miracle.(Lewis [1986b], p. 

49)6 

Lewis calls this the asymmetry of miracles and it is this that accounts for the temporal 

asymmetry of counterfactual dependence in the standard context.  

As Jonathan Bennett ([1984]) notes, there is a problem with this reasoning. Consider 

the actual world where Nixon fails to press the button. Since the laws are Newtonian, they 

are time-reversal invariant (where a set of laws are time-reversal invariant iff if a sequence of 

events from t0 to t1 is allowed by the laws, then the reversed sequence of events t1 to t0 is 

allowed by the laws). Given this, take the state of nuclear-holocaust world just after the small 

miracle causing Nixon to push the button and extrapolate this state forwards and backwards 

in accordance with the laws. Call this world w2. It perfectly matches nuclear-holocaust world 

from the button-pushing and in the future, its dynamical laws are identical to those of the 

                                                 
6 In this passage Lewis is considering re-convergence worlds, but the point is the same.  



 8

actual world and are never violated. w2 is a Bennett world. Bennett notes that whereas the 

miracle in nuclear-holocaust world is a divergence miracle from the actual world’s perspective, 

it is a convergence miracle from w2’s perspective. Thus, it is not the case that every world obeys 

the asymmetry of miracles. Some worlds, like w2, are such that a small miracle can produce 

convergence to them (see Fig. 1). 

From @’s perspective the miracle at w1 is a miracle of divergence, whereas from w2’s 

perspective the miracle at w1 is a miracle of convergence. Nevertheless, @ and w2 have the same 

laws. Thus, there is small-miracle convergence to a world with our laws. So it doesn’t seem 

that there is an asymmetry of miracles: there are worlds with the same laws as our world, and 

yet where there is small-miracle convergence.  

 

Figure 1: Lines represent worlds, with later times towards the top 
of the diagram. Four-pointed stars represent miracles. Anywhere 
that a world’s line is straight, the fundamental laws of @ are being 
obeyed. The actual world is @; w1 is like nuclear-holocaust world; 
w2 is a Bennett world; w3 is like different-past world. 

 

Lewis answers: 

Same laws are not enough. If there are de facto asymmetries of time, not written into the 

laws, they could be just what it takes to make the difference between a world to which 



 9

the asymmetry of miracles applies and a world to which it does not; that is, between a 

world like [@] (or ours) to which convergence is difficult and a Bennett world to 

which convergence is easy. (Lewis [1986b], p. 57) 

Lewis claims that w2 lacks de facto asymmetries of time that @ possesses. This accounts for 

the possibility that a small miracle converges w1 to w2, and rules out worlds like w3 that 

exhibit small miracle convergences to @. We might be able to get easy convergence to a 

world with our laws, but the world to which we are converging won’t exhibit the de facto 

asymmetries true of our world and therefore won’t be as similar to @ as worlds like w1 that 

diverge from @. 

We might wonder, however, what de facto asymmetry guarantees that there will be no 

small-miracle convergence to the actual world. Lewis tells us that it is the asymmetry of 

overdetermination. Every fact, Lewis says, ‘has at least one determinate: a minimal set of 

conditions jointly sufficient, given the laws of nature, for the fact in question.’ (Lewis 

[1986b], p. 49) If a certain fact has more than one determinate at a time, then it is 

overdetermined. If the determinate comes before the fact in question, the fact is 

predetermined, if it comes after, the fact is postdetermined. Our world, Lewis claims, is one 

where facts are over-postdetermined. That is, there are many determinates of a certain fact, 

after it has happened, each sufficient to determine that fact. However, there are much fewer 

determinates before it has happened. Since we are assuming deterministic Newtonian laws, the 

complete set of conditions from any instant of time is sufficient for any fact. Thus, in 

deterministic worlds, every fact has at least one pre-determinate and one post-determinate: 

the complete determinate. But since our world is one with an asymmetry in favor of post-

determinates, this means there must be more of these sufficient sets after the fact than there 

are before the fact.  



 10

This is supposed to underwrite the asymmetry of miracles as follows. If a world, @, 

exhibits the asymmetry of miracles, then a small miracle can produce divergence from @, 

but not convergence to it. Assume that @ exhibits over post-determination, and imagine 

that we are trying to get a world, w3, that converges to @ via a small miracle at t. This means 

that at t, w3 and @ match perfectly, even though they do not match before t. Since @ has 

over post-determination, there are lots of determinates after t of all the things that happened 

in @ before t. Since w3 matches @ after t, w3 must have all these same determinates after t. 

But, of course, some of the things that the determinates speak of did not actually happen in 

w3: there are many “fake” determinates in w3 that seem to entail “facts” that didn’t actually 

happen. Every such fake determinate requires a miracle, and so a convergence world will 

require many miracles (or one big one), rather than one small miracle. So, Lewis claims, if a 

world exhibited the de facto asymmetry of overdetermination, then there is a reason for it to 

exhibit the asymmetry of miracles. It is the asymmetry of overdetermination that makes our 

world unlike a Bennett world.  

Whatever one thinks of how the asymmetry of overdetermination is meant to 

ground the asymmetry of miracles, the point is moot: it is false that our world exhibits an 

asymmetry of overdetermination. To see why, consider an example that Lewis gives of such 

overdetermination: a spherical wave expanding from a point source to infinity. This happens 

in our world, says Lewis, but the opposite (a spherical wave contracting from infinity to a 

point source) never does. He writes:  

A process of either sort exhibits extreme overdetermination in one direction. 

Countless tiny samples of the wave each determine what happens at the space-time 

point where the wave is emitted or absorbed. (Lewis [1986b], p. 50)  



 11

If this were true, then any condition describing a small sample of the wave after it is emitted 

is a determinate for the emission of the wave from a point source. Since there are many such 

small samples of the wave after the point of emission, but not before, we would have 

extreme overdetermination.  

Lewis is certainly right that at our world waves do expand to infinity and never 

contract from infinity, but this isn’t an example of overdetermination. Recall, a determinate 

is sufficient, together with the laws of nature, for the fact in question. A set of propositions 

about a small portion of the wave, however, is not sufficient for its emission from a point. 

To get sufficiency  we must add the further information about what is happening outside 

this region. Perhaps, for example, the small part of the wave is not a part of a spherical wave 

at all, but merely a part of space that is identical to what this part of the wave would be like, 

were there a wave. What is sufficient for inferring the point source emission is not some 

small part of the wave but the complete set of conditions at any time after the fact in 

question. Such a set is the biggest possible determinate for the fact in question. If the biggest 

possible determinate is also the minimal determinate, then there is only one determinate for 

this fact. Thus, there is no overdetermination in either direction, neither past nor future.7  

Lewis’s account is now faced with a problem: The asymmetry of overdetermination 

would ensure that miracles of convergence to our world are larger than miracles of 

divergence. This, plus the similarity metric, would give us the asymmetry of counterfactual 

dependence. But there is no asymmetry of overdetermination at our world. So we are left 

with no assurance that the asymmetry of miracles is true of our world, and thus no assurance 

that Lewis’s account will yield the desired asymmetry of counterfactual dependence.  

                                                 
7 After writing this paper, it was brought to my attention that Sanford ([1989], p. 187) makes a similar point. 



 12

It is important to emphasize that the claim that the asymmetry of overdetermination 

is false should not be confused with the claim that there are no de facto asymmetries true of 

the actual world. There do seem to be de facto asymmetries at the actual world. One of these 

is a kind of epistemic asymmetry of overdetermination. It is true that there are many more 

conditions after an event, rather than before an event, which we are able to notice and that 

make it reasonable for us to infer that some event occurred. But this asymmetry isn’t written 

into the fundamental laws. Since the similarity metric is only sensitive to fundamental laws, 

an epistemic asymmetry isn’t capable of underwriting the asymmetry of miracles.8 

Given all this, we have no assurance that the asymmetry of miracles is true of the 

actual world. If it is not, then there is nothing in Lewis’s similarity metric that will ensure we 

get the asymmetry of counterfactual dependence. Lewis’s account is primed for a 

counterexample. Adam Elga delivers one. 

 

3 Elga’s Counterexample 

Here is Adam Elga’s ([2000]) counterexample. In the actual world (@), Gretta cracks an egg 

at 8:00 am letting it drop onto the hot frying pan in her kitchen.  Now, consider: 

(Egg) If Gretta hadn’t cracked the egg, then at 8:05 there wouldn’t have been a cooked egg on the 

pan. (Elga [2000], p. S314.) 

If (Egg) is to come out true, then the closest no-egg-cracking world must be a world where 

there isn’t a cooked egg on the pan. This world perfectly matches @ until just before 8:00, at 

which point there is a small miracle keeping Gretta from cracking the egg. After this, the 

laws of @ are obeyed. Call this world w1. 

                                                 
8 We might ask ourselves why there is an epistemic asymmetry overdetermination. It seems that any such 
epistemic asymmetry of overdetermination must be grounded in some physical asymmetry. At the end of this 
paper, I propose one such physical asymmetry. 



 13

 If the Lewisian strategy is to succeed, we must rule out a competing world, w3. 

Before 8:05, w3 does not match @. In this world the laws dictate that no egg is cracked. 

However, just before 8:05 there is a small miracle that converges w3 into perfect match with 

@ at all later times. If there is such a world, (Egg) is not true.  

Lewis claims that there is no w3 where a small miracle leads to convergence with @. 

Elga, however, gives us a recipe for constructing w3. We start with the state of @, S1, at 8:05 

just after the egg is cooked, and extrapolate things forward according to the laws. Thus, w3 

matches @ from 8:05 into the future. To construct the before-8:05 section of w3, we take the 

velocity-reverse of state S1 (call it Z1) and extrapolate backwards according to the laws—but 

not before making a small miracle change. 

Since the process of the egg cooking (from 8:00 to 8:05 in @) is an entropy-

increasing process, the reversed process (from 8:05 to 8:00) is an entropy-decreasing process. 

Now, consider the phase space of the kitchen system, and the subregion of this phase space 

corresponding to the situation at 8:05: a cooked egg sitting in a frying pan. We know from 

classical statistical mechanics that the volume of states in this subregion with entropy-

decreasing futures (states like Z1) is unbelievably small compared to the volume of the rest of 

the subregion. Elga concludes that a small miracle moves us from the extremely rare state, 

Z1, to a more typical state with an entropy-increasing path. Thus, a small miracle at 8:05 

results in normal entropic behavior in the backwards time direction. If we run time forward, 

we see something strange. At 8:00 in w3 there is a cooled down and cooked egg in the pan. 

As we move towards 8:05 the cooked egg spontaneously heats up, as does the pan until it 

reaches a just-cooked state. Suddenly, there is a small-miracle change bringing w3 to state S1. 

After this the dynamical laws take over ensuring that w3 perfectly matches the actual world. 

The egg was never cracked, and yet it is cooked. 



 14

 

4 Possible Solutions 

We are now in position to consider solutions that can be made on behalf of Lewis. I will 

consider solutions that reflect the strategy that Lewis himself articulated in response to 

Elga’s counterexample: 

[…] the worlds that converge onto worlds like ours are worlds with counter-entropic 

funny-business. I think the remedy—which doesn’t undercut what I’m trying to 

do—is to say that such funny business, though not miraculous, makes for 

dissimilarity in the same way miracles do. (quoted in Bennett [2003], p. 296) 

There are several ways to take this comment by Lewis. One option is to construe the 

counter-entropic funny-business not as violations of law, but rather as violations of kinds of 

particular facts. To this end one might invoke criterion 4 of the similarity metric. A different 

option is to try to construe some law of thermodynamics or statistical mechanics as a 

fundamental law and thus have the counter-entropic funny-business literally be a miraculous 

violation of such a law. To this end one might attempt to show that something like David 

Albert’s ([2000]) Past Hypothesis could be a Lewisian fundamental law. A different strategy 

is to try to construe some law of thermodynamics or statistical mechanics as a non-fundamental 

or special science law, and thus have the counter-entropic funny-business be a non-

fundamental miracle. It is this third option that I will pursue.  

 

5 Structure of the Special Science Solution 

Call this third option, the special science solution. The guiding idea behind this solution is that 

thermodynamically reversed worlds violate laws of the actual world. However, these violated 

laws are not fundamental laws. Instead, Elga’s w3 will be seen to violate non-fundamental or 



 15

special science laws. In what follows I will show how one can see thermodynamics as a 

special science, and thus countenance the Second Law of Thermodynamics as a special 

science law. If violation of such special science laws is built into the similarity metric, then 

there is a way out of Elga’s argument.9 

First, however, we need to say something about thermodynamics and particularly the 

Second Law. Thermodynamics is a science chiefly concerned with macroscopically 

observable properties of systems. Thermodynamics formalizes the relationship between 

properties such as temperature, pressure, and volume for macroscopic systems. An 

introductory text on the topic puts it as follows: 

[Thermodynamics] deals broadly with the conservation and interconversion of 

various forms of energy, and the relationships between energy and the changes in 

properties of matter. The concepts of thermodynamics are based on empirical 

observations of the macroscopic properties of matter in physical, chemical, and 

biological changes, and the resulting observations are expressed in relatively simple 

mathematical functions. (Gokcen & Reddy [1996], p. 1)10 

                                                 
9  Of course, by adding consideration of all special sciences to the similarity metric (and not just 
thermodynamics) we’ll probably have more than we need to solve Elga’s problem. Adding thermodynamics on 
its own may be enough to get the time asymmetry we’re after. So why add in all the special science laws? The 
main reason for adding in all the special science laws is that in the standard context the similarity that special 
science laws add to a world matter for the evaluation of counterfactuals. For instance, we seem to think that the 
following is true: 

If the apple farmers’ crop yield had outrun demand, the price of apples would have gone down.  
Now, if we go to worlds where the antecedent is made true by a small fundamental miracle and where there are 
no other violations of fundamental law or thermodynamic laws, we might only be left with worlds where the 
economic laws hold. If the fundamental laws together with the thermodynamic laws entail that the special 
science laws stay fixed then this is certainly the case, and so putting them in the similarity metric is superfluous 
(though it doesn’t do any harm). But this might not be the case. And if the thermodynamic laws and the 
fundamental laws aren’t enough to entail the special science laws, then it seems to be a good thing to have them 
in our similarity metric. Having the special science laws (and not just the thermodynamic laws) in our similarity 
metric does not add anything with respect to Elga’s problem. Rather, the idea is that if we add them in to the 
similarity metric—and it might be nice to do so for reasons just given—then Elga’s problem can be taken care 
of, too. Thanks to an anonymous referee for emphasizing this point. 
10 Another text puts it: ‘The task of thermodynamics is to define appropriate physical quantities (the state 
quantities), which characterize macroscopic properties of matter, the so-called macrostate, in a way which is as 



 16

The Second Law of Thermodynamics is one of the axioms of this scientific theory. Rudolf 

Clausius formulates it as follows: ‘Heat can never, of itself, flow from a lower to a higher 

temperature.’ (Burshtein [1996], p. 277)  The Second Law receives a more modern 

formulation using the concept of entropy.11 Using this, the Second Law tells us that in a 

spontaneous evolution of a thermally closed system, the entropy never decreases.12  

 The before-miracle section of w3 violates the Second Law. Specifically, consider the 

(approximately) closed system of the kitchen containing the frying pan and the egg. As we 

move forward in time before the small miracle, the pan spontaneously heats up while the egg 

spontaneously gets warmer and less decayed. For this to happen, energy is dispersed from 

the rest of the kitchen and into the frying pan and egg. As the frying pan and egg grow 

warmer there is spontaneous energy dispersal from the cooler surroundings to the warmer 

frying pan and egg. This is in direct conflict with Clausius’s formulation of the Second Law, 

but also with the entropy-version of the Second Law. The dispersal of energy from warmer 

bodies to cooler bodies in a closed system results in a net increase of entropy. Thus, in our 

kitchen system, there is a net decrease of entropy since there is dispersal of energy from the 

cooler bodies (the rest of the kitchen) to the warmer bodies (the pan and egg). Now, the 

kitchen is not an entirely closed system, but as long as there is no offsetting increase in 

                                                                                                                                                 

unambiguous as possible, and to relate these quantities by means of universally valid equations (the equations of 
state and the laws of thermodynamics).’ (Greiner, et al [1995], p. 3). Callender ([2006]) offers us the following: 
‘Thermodynamics is a ‘phenomenal’ science, in the sense that the variables of the science range over 
macroscopic parameters such as temperature and volume. Whether the microphysics underlying these variables 
are motive atoms in the void or an imponderable fluid is largely irrelevant to this science.’ 
11 In thermodynamics, the entropy of a system A is usually defined as the integral of dQ/T from some 
(arbitrarily picked) state B to A of a reversible process, where Q is the energy of the system and T is the 
temperature. It is important to note that it is a contentious point whether or not the concept of entropy used in 
classic thermodynamics is the same as the concept of entropy used in statistical mechanics. The relation 
between statistical mechanics and thermodynamics is interesting, contentious, and difficult to determine. I do 
not make claims about that relation here. An extremely tentative view about the relation between the two is 
that statistical mechanics is an attempt to explain how we get the special science laws of thermodynamics, given 
certain fundamental physical laws. It is important to note that I am attempting to construe classical 
thermodynamics as a special science, not statistical mechanics.  
12 See Callender ([2001]); Buckingham ([1964], p. 28); Craig ([1992], p. 39); Lieb & Yngvason ([2000]). 



 17

entropy outside the kitchen (as there would be if, say, we were considering a refrigerator as 

our approximately closed system), then we will have a violation of the Second Law. Further, 

I follow Elga in assuming that not only will the small miracle put the system onto a 

backwards-directed entropic path, but this backwards-directed increase in entropy will 

spread out infecting a larger and larger area as time goes backwards. Here is how Elga 

describes w3: ‘In the distant past, the infected region is huge. Within that region are events 

that look thermodynamically reversed. Events outside of the infected region look 

thermodynamically typical.’ (Elga [2000], p. S323) Since the entire world is a closed system, 

we have here a violation of the Second Law. Thus, if the Second Law is somehow worked 

into the similarity metric, w3 is less similar to @ than w1. 

One could attempt to construe the Second Law as a fundamental law and work it 

into the similarity metric in this way. This option is problematic.13 A Lewisian fundamental 

law is a theorem in the best deductive systematization of the phenomena. The best system 

has the best balance of simplicity and strength. Importantly, however, the theorems must 

reference only perfectly natural properties. If there is no such restriction, then the constraint 

of simplicity is rendered trivial. The problem with construing the Second Law as 

fundamental is that the predicate ‘entropy’ does not seem to refer to a perfectly natural 

property. 14 The sharing of a certain amount of entropy between two systems doesn’t make 

for similarity in the way that the sharing of perfectly natural properties is supposed to make 

for similarity. The Second Law would be very complex if formulated by referring only to 

                                                 
13 This problem with candidate Lewisian fundamental laws is given in Schaffer ([2007]). 
14 The predicate ‘entropy’ refers to an organizational property that complex systems can have. Take two 
systems with parts that are intrinsic duplicates of each other and that both instantiate the property of having 
low entropy. These systems need not be perfect duplicates of each other. According to Lewis, though, things 
that have exactly the same perfectly natural properties are perfect duplicates of one another. See for instance, 
Lewis [1986a], p. 61. 



 18

perfectly natural properties. Thus it seems unlikely the Second Law is eligible to be a 

fundamental law of nature.15 

One might think that there is a different way of getting thermodynamics into the 

similarity metric. David Albert ([2000]) and Barry Loewer ([2007], [forthcoming]), for example, 

give a different kind of picture where the fundamental laws entail the special science laws and 

the thermodynamic regularities that we observe. However, on this picture, the fundamental 

laws are somewhat different than expected, consisting of Newton’s Laws, the Past 

Hypothesis—which posits that the universe began in a state of low entropy—and a statistical 

mechanical probability distribution over initial conditions. This is very interesting proposal, 

and would allow for a different response to Elga’s example. However, let me note four 

reservations with such an account. First, the Past Hypothesis and the probability distribution 

are not regularities. Lewis ([1999], p. 41) claims that only the regularities in a best system will 

be laws. Second, if the predicate ‘low entropy’ does not refer to a perfectly natural property, 

then the Past Hypothesis does not qualify as a fundamental law. Third, it is an article of faith 

that these three fundamental laws will entail all the special science laws. Certainly we’d expect 

the special science laws to be roughly consistent with the fundamental laws, but it is a bold 

claim that they are entailed by the fundamental laws. Finally, it is important to note that on a 

natural way of extending the Albert/Loewer account so that it gives truth-conditions for 

counterfactuals (see Loewer [2007]), counterfactuals are never fully true or false, but instead 

have probabilities associated with them. The solution I will advocate does not have this 

consequence. Although there are unlikely possibilities where a small miracle puts us on an 

anti-entropic course in the future (and so even though Nixon did press the button, no nuclear 

                                                 
15 To avoid this problem, one could argue that the predicates referenced by thermodynamics in fact are 
perfectly natural. A full treatment of this is beyond the scope of this paper. Nevertheless, note that the 
suggestion moves away from Lewis’s conception of perfectly natural properties and so presents a prima facie 
difficulty.  



 19

holocaust resulted), such worlds do not compete with the well-behaved worlds because they 

violate the Second Law. I do not think that any of these points motivate a rejection of the 

Albert/Loewer picture. However, I think they do motivate an exploration of a different 

Lewisian solution.   

Here, then, is the structure of such a solution. The basic idea is that we work the 

Second Law into the similarity metric by adding an extra criterion. If Lewis’s first three 

criteria leave us with worlds that are tied for similarity, we turn to the new criterion: 

 (3.5) It is of the fourth importance to avoid violation of special science laws.  

Lewis’s criterion 4, which does little work for Lewis, is then moved to fifth importance.16 

Worlds that violate special science laws are thus less similar to the actual world than those 

that do not. For this special science solution to be successful, several things must be shown: 

(i) it must be shown that thermodynamics can be thought of as a special science, and (ii) it 

must be shown how special science laws could sit with Lewis’s view of lawhood. I will 

address both of these shortly. But first note how this new criterion in the similarity metric 

would block Elga’s counterexample: w1 and w3 are tied for similarity with respect to criteria 1-

3, so we turn to criterion 3.5. If thermodynamics is a special science, then the Second Law is 

a special science law, so the fact that w3 violates it, but w1 doesn’t counts against w3. Thus, w1 

is more similar to @ than w3 and the counterexample is avoided.  

But note that even before addressing points (i) and (ii) things are not so simple as 

this.17 For the proposed solution assumes that worlds like w3 and more well-behaved worlds 

like w1 will always be tied for similarity up through the first three criteria. But this may not be 

the case. Say that Gretta’s egg-cracking occurs very early in @. Thus, there is a relatively 

short spatiotemporal region before Gretta cracks the egg and relatively long spatiotemporal 

                                                 
16 In order of importance, then, the criteria are ordered: (1), (2), (3), (3.5), (4). 
17 Thanks to an anonymous referee for suggesting this objection. 



 20

region after she cracks the egg. Now imagine the w1 and w3 worlds relative to such an @.  

Both will be tied with respect to criterion 1 since neither w1 nor w3 exhibit big violations of 

fundamental law. However, w3 would seem to be favored over w1 with respect to criterion 2. 

For w1 has a relatively short region of perfect match with @ while w3 has a relatively long 

region of perfect match. Given this, w3 is more similar to @ than w1, and we never have a 

chance to appeal to criterion 3.5. So, it would seem, the special science solution doesn’t solve 

all Elga-style counterexamples, only counterfactuals with antecedents that aren’t relatively 

early in @’s history. This seems like a problem. 

 It is useful in working towards a solution to note Lewis’s primary purpose for 

criterion 2. He uses criterion 2 to rule out worlds like w2 (see fig. 1). But to rule out such 

worlds, we do not need to quantitatively maximize the region of perfect match. Rather, we 

only need to require that there is some region of perfect match. w2 fails in this respect, so it is 

ruled out of contention for being most similar to @. So, it seems that we could reinterpret 

criterion 2. Instead of instructing us to quantitatively maximize the region of perfect match, 

it is to be modified so that any future perfect match counts for the same as any past perfect 

match no matter the size of the region of perfect match. 

 On its own, such a modification is probably inadequate. For criterion 2 may play a 

secondary role with regard to how early or late a miracle occurs to bring about the truth of 

the antecedent of the counterfactual. Consider a counterfactual with the antecedent, ‘If my 

alarm hadn’t gone off this morning…’. Now imagine two worlds, world m where there is a 

small miracle that causes the clock to malfunction, and world s where there is a small miracle 

the night before that keeps me from setting the alarm. One world has a slightly longer region 

of perfect match than the other, but according to the suggestion above this does not render 

m more similar than s. Addressing the general question of which world should be closer to get 



 21

the right truth conditions for various counterfactuals would go beyond the scope of this 

paper.18 But it may be desirable to appeal to maximization of spatiotemporal match in such 

situations to decide the issue of which world is closer to the actual world. The solution is to 

note that the second criterion can be formulated so that quantitative comparisons do matter 

when we’re considering worlds with different amount of past match, or different amounts of 

future match, but does not matter when comparing a world with one amount of future 

match and a different world with a different amount of past match. So, quantitative 

comparison between two worlds with differing regions of perfect past match or between two 

worlds with different perfect future match does matter for similarity. But we do not 

quantitatively compare a region of past match with a region of future match. Importantly, 

this does not stipulate any time asymmetry. It treats the past as something different than the 

future, but it does not treat the past differently from the way it treats the future. So, no 

asymmetry of time is put in by fiat in a way that would be unacceptable. Adopting this 

modification will have the effect that even an @ with an early egg-cracking is a world where 

w1 and w3 are tied for similarity for criteria 1-3. This then allows criterion 3.5 to decide all 

such Elga counterexamples as desired.19 

 

6 The Special Science Solution 

                                                 
18 For some thoughts about the complexity of such considerations, see Bennett ([2003], pp. 209-221 ).  
19 One might be worried that there is no intuitive support for taking this modification of criterion 2 to be 
getting at similarity any better than the original criterion 2. There is a sense in which this is correct. However, in 
this paper I am explicitly adopting Lewis’s methodology with respect to similarity metrics. Such metrics, Lewis 
notes, need not be intuitive in themselves. Rather, the intuitiveness of a similarity metric is to be judged in what 
it says about the truth-conditions of counterfactuals. As Lewis writes in his ([1986b]): 

[…] we must use what we know about the truth and falsity of counterfactuals to see if we can find 
some sort of similarity relation—not necessarily the first one that springs to mind—that combines 
with Analysis 2 to yield the proper truth conditions. …we must use what we know about 
counterfactuals to find out the appropriate similarity relation—not the other way around. (p. 43)  

‘Analysis 2’ refers to Lewis’s well-known account of the truth-conditions of counterfactuals that I set out at the 
start of the paper. 



 22

With the structure of the special science solution explained, I will address points (i) and (ii). 

First, I will show that thermodynamics can be thought of as a special science, and then I will 

show how special science laws could be accounted for given Lewis’s view of lawhood. After 

this I will consider several objections to the proposal.   

 

6.1 Thermodynamics as a Special Science 

Thermodynamics isn’t among the paradigm examples of special sciences. But it seems that 

any plausible way of articulating which features make for a special science will put 

thermodynamics in with the others.   

 One key feature of a special science is the following: special science laws hold 

specifically at their own “level” of inquiry. This idea can be made more perspicuous. 

Fundamental laws hold between the entities of fundamental physics, even when we consider 

these entities singly.20 For instance, the fundamental laws of motion hold between the 

entities of fundamental physics. These laws of motion, of course, also hold between entities 

of the special sciences – entities like molecules, cells, brain states, and monetary units. A 

feature of the special sciences is that their laws do not have this wide applicability. Laws of 

supply and demand simply do not apply to the entities of fundamental physics taken singly. 

Instead, we need massive pluralities of specially arranged fundamental entities before the 

laws of supply and demand apply. This is one feature of special science laws. If we consider 

this feature, thermodynamics looks like a special science. Thermodynamic laws invoke 

properties like pressure, temperature, and entropy, and macroscopic entities like adiabatic 

                                                 
20 What is a fundamental entity of physics? Following Lewis, we can think of the fundamental entities as those 
things—whatever they may be—that form the ultimate parts of the Humean mosaic. There is a worry here, 
that the entities of fundamental physics might not be single particles, but actually massive pluralities of 
quantum-entangled particles. This possibility is set aside for two reasons. First, we are considering classical 
correlates of the actual world. Second, it is not clear that if this possibility damages any of what is said. 



 23

systems. Laws invoking such entities and properties do not apply to fundamental entities 

taken singly. 

 A second key feature of special sciences is that they are blind, in a certain sense, to 

what takes place at the fundamental level. In economics, for example, we have laws about 

supply and demand and as long as we have something that plays the monetary role (and 

several others), these laws hold. This is true even if fundamental physics were such that 

collections of different fundamental entities, related in different ways, combined to play that 

role. Imagine we find that fundamental physics is terribly mistaken. There is a clear sense in 

which it doesn’t matter to economic laws what our fundamental science looks like, as long as 

the monetary role (among others) is filled by something. Similar comments could be made 

about psychological laws and biological laws. If this is a mark of the special sciences, then we 

again have reason to think that thermodynamics is a special science. The laws of classical 

thermodynamics are relations between things like pressure, temperature, and entropy. It 

doesn’t matter to thermodynamics whether the systems we are discussing are composed of 

the fundamental particles of physics so long as the systems have something that play the 

proper roles. In fact, Lieb & Yngvason ([2000]) derive the Second Law in a way that is 

completely blind to what the various states are like at the fundamental physical level.21 Thus, 

though thermodynamics is not traditionally listed as a special science, it seems that it should 

be.22 

 

 

                                                 
21 Also, Greiner, et al write: ‘…classical macroscopic thermodynamics…is of great importance: the concepts of 
thermodynamics are very general and to a great extent independent of special physical models, so that they are 
applicable in many fields of physics and the technical sciences.’ ([1995], p. 3). 
22 Callender ([1997]) also claims that thermodynamics should be thought of as a special science, though his 
reasons are different than mine. 



 24

6.2 Lewisian Special Science Laws 

The next task is to show how special sciences can sit within a Lewisian framework of 

lawhood. Recall that for Lewis a fundamental law is a theorem (referencing only perfectly 

natural properties) in the deductive system that is the best balance of simplicity and strength. 

I propose that a Lewisian special science be thought of as a science that studies a certain 

subset of all the phenomena and a special science law is a theorem in the best systemization 

of these phenomena when we limit ourselves to the vocabulary of a special science.  For example, 

special science laws in psychology are the regularities in the best system that is formulated 

using psychological vocabulary. These terms do not refer to perfectly natural properties, and 

so psychological laws are not fundamental. But they are kinds of laws nonetheless. 

There is an immediate worry: according to this account we describe the world using a 

certain vocabulary. Then we systematize these descriptions in the best way possible. The 

regularities are our special science laws. But where does this vocabulary come from? It is 

tempting to claim that any vocabulary is on the table. Describe the world in some 

vocabulary. If that description can be strongly and simply systematized, then we have some 

special science laws. This, however, will not work. If we choose bizarre enough vocabularies, 

then we can get any kinds of laws we want. This is not only a problem because such a move 

would allow us to construct bizarre laws. The primary problem is that by constructing the 

predicates in the right way, we would be able to make any world violate or agree on as many 

special science laws of any other world as we want. This would render criterion (3.5) of the 

similarity metric irrelevant. 

The solution to this difficulty is to restrict the vocabularies that can be used. In the 

fundamental case, Lewis limits the vocabulary to those that reference only perfectly natural 

properties. In this case, one should say that the vocabularies can only reference imperfectly 



 25

natural properties, those properties the having of which make for similarity. Note that there 

are two conceptions of imperfectly natural properties. One corresponds to those properties 

that can be built up out of logical constructions of perfectly natural properties. On this 

conception, every non-perfectly-natural property is imperfectly natural.  The other 

corresponds to what Lewis needs in his response to Putnam’s paradox:  

[…] the realism that recognizes a nontrivial enterprise of discovering the truth about 

the world needs the traditional realism that recognizes objective sameness and 

difference, joints in the world, discriminatory classifications not of our own making 

[…]. What it takes to solve Putnam’s paradox is an objective inegalitarianism of 

classifications, in which grue things (or worse) are not all of a kind in the same way 

that bosons, or spheres, or bits of gold, or books are all of a kind. (Lewis [1999], p. 

67) 

This conception is distinct from the first conception. We can see this by noting that the 

property of being grue is more natural than the property of being a book when we rate 

things in terms of constructions from perfectly natural properties, and yet Lewis says that 

grueness is less natural. It is this second notion of imperfect naturalness that is needed to 

ground special science laws. Lewis already needs and uses such a notion of imperfect 

naturalness for his philosophy of language (Lewis ([1986a], p. 61, [1999], pp. 13, 49, and 66). 

Further, this notion of naturalness is not in any way subjective. Lewis appeals to such a 

notion of naturalness to solve problems of indeterminacy of reference. On pain of 

circularity, then, it is not the fact that we refer to these properties that make them imperfectly 

natural. Rather, it is their imperfect naturalness that make them eligible referents.    

In a recent paper, John Hawthorne speaks favorably about this conception of 

imperfect naturalness. He writes: ‘We should thus be willing to give relative naturalness a life 



 26

of its own, one that allows properties that are of equal definitional distance from the 

microphysical ground floor to be of radically unequal naturalness.’ ([2007], p. 434) And later, 

using this notion of imperfect naturalness to get a handle on what he calls ‘semantic 

properties’—the eligible referents of our words—he writes: 

[…] once we have relinquished the idea that relative naturalness is to be tied to ease 

of definability from the ground floor, we are free to take a more elevated view of 

semantic properties and relations themselves. In particular, we should take seriously 

the idea that while semantic properties and relations don’t occur at the ground floor, 

they are very natural, not gerrymandered. […] Granted, there is no recipe for 

generating the semantic properties from the fundamental ones. But given that 

naturalness does not have to be tied to ease of definability, that does not indict the 

naturalness and importance of semantic properties. ([2007], p. 435)23 

This notion of imperfect naturalness is a respectable one, and it is one that is needed 

elsewhere in Lewis’s system. The virtue of such a notion rests in the uses to which it can be 

put. It can and should be used here.24  

So, with our vocabulary restricted to referencing the imperfectly natural properties 

we go about describing the world and trying to uncover regularities. Certain subsets of the 

phenomena will permit of simple and strong systematization. The regularities of such 

systems are the special science laws. In practice, of course, the process of coming up with 

special science laws is intimately related with the process of correctly formulating the 

                                                 
23 For more on this distinction, see Schaffer ([2004]). Lange ([2000b]) also discusses something like this notion 
of  imperfectly natural properties (see, especially, pp. 215-216), but he seems to tie this notion of naturalness to 
our practices in a way that I do not. 
24 It is worth noting that we could rank the imperfectly natural properties in terms of their degree of 
naturalness, if we desired, via their definability in terms of the perfectly natural properties. In this way, for 
example, the properties of chemistry might be more natural than the properties of biology. Importantly, 
however, the properties of both these sciences are members of the privileged class of imperfectly natural 
properties. 



 27

vocabulary to be used. Certain choices of vocabulary simply won’t yield a best system that is 

good enough: the best system that results is one with no regularities, or one that is terribly 

uninformative. The actual process is one of mutual refinement, in which the formulation of 

laws and the vocabulary used affect each other. Nevertheless, as a characterization of what a 

special science law is, we can say that a special science law is a regularity in the best system 

formulated when restricting oneself to the vocabulary of a special science, where a 

vocabulary of a special science is one that refers to imperfectly natural properties. 

There is a remaining worry, however, concerning the fact that special science laws 

usually have exceptions. The picture I have sketched doesn’t seem to allow for such 

exceptions. Though incredibly unlikely, it might be true that in the actual world there are never 

exceptions to the Second Law.25 Nevertheless, a general account of special science laws, 

should show how such laws could have exceptions.  

One way to think about exceptions follows Jerry Fodor ([1974]). Here is his familiar 

picture. We have a special science law: 

SS-Law: Sx � S*x26  

‘S’ and ‘S*’ are predicates of a special science, so they refer to properties that are not 

perfectly natural. Thus, we can formulate all the ways to be S and all the ways to be S* in 

terms of perfectly natural properties: 

 Sx: P1x or P2x or … or Pnx  

S*x: P*1x or P*2x or … or P*mx 

Each of the Pis and P*is can be thought of as an abbreviation for a longer specification of 

the pattern of instantiation of perfectly natural properties. One might think that there are 

                                                 
25 Of course, this feature might make one doubt that the Second Law is a special science law. I don’t think this 
is right. Exceptions are typical of special science laws, but they need not be necessary. 
26 Fodor, it seems, means to read the ‘�’ in a metaphysically robust sense. I mean no such thing. The ‘�’ is 
just shorthand for ‘It is a law that if…then…’. 



 28

fundamental laws entailing that each Pix evolves to some P*ix. However, if the SS-Law is not 

entailed by the fundamental laws, then there will be some Pix that does not evolve to some 

P*ix. This would not necessarily result in an exception to the SS-Law unless there actually is 

some x that instantiates some Pi that does not evolve to P*i. But when there is, we get an 

exception to the SS-Law. Perhaps there is some instantiation of perfectly natural properties 

by a, Pna, which is also an instantiation of S-ness. However, Pna evolves by fundamental law 

to some state P′a, where P′a is not identical to any P*ia. Given this, something that is S does 

not become S* and we have an exception to our SS-Law.  

According to this picture of exceptions, the special science laws need not be 

consistent with the fundamental laws. The special science law says that S will become S*, but 

it does not. How, one might ask, can this be? The right thing to say is that it might be that no 

imperfectly natural vocabulary yields perfect regularities, but some such vocabulary gives us 

approximate regularities. Thus, the best systematization of the phenomena described in 

some imperfectly natural vocabulary is one that doesn’t give us perfect regularities, but only 

approximate regularities. But this is not catastrophic. A certain special science law might 

come out as a theorem in the best system for whatever vocabulary the special science is 

conducted in, even if the special science law is not strictly consistent with the fundamental 

laws and so has exceptions. It is a virtue of best system view of laws that there can be laws 

with exceptions.27  The gains in simplicity could make it worth adopting a special science law 

not strictly consistent with the fundamental laws.28   

                                                 
27 In his presentation of the best system account of fundamental laws, Lewis holds that fundamental laws are 
exceptionless. This is not, however, a consequence of the best system view of laws, but rather an additional 
constraint on fundamental laws adopted by Lewis. 
28 Obviously this will be a balancing act. If there are too many exceptions no matter how we formulate things, 
we simply reject the idea that there are any special science laws for the phenomena picked out by the 
vocabularies in question. 



 29

This concludes my sketch of how special science laws can fit within a Lewisian 

picture of laws. I have also argued that thermodynamics should be treated like a special 

science, just as we treat chemistry, biology, psychology, or economics. Given all this, the 

hope is that one special science law that is true of the actual world is the Second Law of 

Thermodynamics. It unifies a vast amount of information about the behavior of many, many 

different macroscopically described systems. Because of the vast number of systems the law 

applies to (in part due to the imperfectly-natural predicate ‘entropy’), the law is extremely 

simple but also very strong. It serves to unify a myriad of different kinds of time-asymmetric 

behavior. 

It is instructive to pause here and consider some recent work by physicists Lieb & 

Yngvason ([2000]) on the foundations of thermodynamics. Their work supports the view 

that the Second Law is a special science law in the Lewisian sense just described. In their 

paper, they give an axiomatic treatment of entropy and, in turn, are able to formulate the 

Second Law in what they believe is its most general form. They start with the relation: 

adiabatic accessibility. A state Y is said to be adiabatically accessible from state X iff Y can be 

reached from X ‘without leaving an imprint on the rest of the universe, apart from the 

displacement of a weight.’ (Lieb & Yngvason, [2000], p. 33) They then give six axioms that 

restrict how this relation must behave. Finally, they make the assumption they call the 

Comparison Hypothesis. This hypothesis stipulates that we have enough pairs of states that 

bear the adiabatic accessibility relation to each other, ensuring that we have enough states 

that are related to each other in the right sorts of ways so as to be able to derive a lawlike 

relation. Given, this, they show how to derive a unique entropy function for every state, and 

a corresponding Second Law that unifies this behavior. Two points deserve note. First, it 

does not matter to the entropy function or to the Second Law, what the various states are 



 30

like at the fundamental physical level. All that is required is that a certain accessibility relation 

holds between the states. This supports the claim that thermodynamics should be seen as a 

special science. Second, note that the Comparison Hypothesis is important because it 

ensures that there are enough of the right sorts of regularities to derive the entropy function 

and the Second Law. So, on this account, the Second Law is straightforwardly a 

systematization of all the data when we restrict ourselves (via the Comparison Hypothesis) 

to certain states and systems. This fits in well with the Lewisian conception of special 

sciences just given. A special science law is a law in the best system, when we restrict 

ourselves to a certain class of phenomena described in the vocabulary of a special science. 

 

7 Objections 

So far I have shown how Thermodynamics may be seen as a special science, and how a 

Lewisian might offer an account of special science laws. Together with the new similarity 

metric, this proposal blocks Elga’s counterexample. However, one might worry that the new 

similarity metric will lead to trouble in other areas.  

 Consider the following objection to the proposed new similarity metric.29 Grant that 

biology is a special science, and imagine that there was some critical event that occurred in 

the past, say a crucial step in the move toward DNA, in spacetime region R, that lead biology 

on its current course. Let’s assume that had this particular critical event not occurred, then 

biology would have been very different. Now, consider the counterfactual: 

(L1) If lightning had struck in region R, then the laws of biology might have been very different.  

(L1) strikes us as true. But one might wonder how the proposed similarity metric can issue 

this result. For imagine two worlds, c and d, tied for similarity according to criteria 1-3. In c, 

                                                 
29 This objection was suggested by an anonymous referee.  



 31

lightning strikes in just the particular region of R to prevent the critical event, and biology is 

radically different. In d, lightning strikes in a different part of region R, the critical event 

occurs, and biology is the same. It looks as if criterion 3.5 will issue the verdict that d is 

closer to the actual world than c in which case the following is true: 

(L2) If lightning had struck in region R, then the laws of biology would have been just as they 

actually are. 

But (L1) and (L2) appear to conflict.  

The solution to this difficulty is to say that (L2) is true, but that there is a reading of 

(L1) on which it is consistent with (L2). There are two ways of reading a counterfactual like 

(L1). Consider: 

(L1-nwn) It is not the case that: if lightning had struck in R, then the biological laws 

would have been just as they actually are. 

This does indeed conflict with (L2). However, consider: 

(L1-wbp) If lightning had struck in R, it would be that: different biological laws are 

possible. 

(L1-wbp) says that all the most-similar lightning strike worlds, are worlds where different 

biological laws are possible. But this is compatible with (L2). So, I opt for (L1-wbp) as a way 

of understanding why (L1) strikes us as true.30 One might object that the (L1-nwn) reading of 

                                                 
30 This kind of response is not novel. Lewis makes essentially this sort of move in the postscript of 
‘Counterfactual Dependence and Time’s Arrow’ ([1986b]). Lewis is worried about the following sorts of cases. 
Assume the world is indeterministic. Then there are all sorts of chancy events happening. But now consider the 
Nixon case. Lewis ruled out the convergence world (different-past world) by saying that it would take a big 
miracle to get perfect convergence to the actual world where there is no nuclear holocaust. But if there are 
chancy events, then these chancy events might do the job of the big miracle on their own, converging different 
past world onto the actual world, without any miracle at all. Lewis’s solution is to appeal to what he calls ‘quasi-
miracles’, which are to be chancy lawful events that are very much like miracles, and detract from similarity in 
the same way. Never mind what one thinks of quasi-miracles. What is important is that Lewis notices a 
problem:  

For if quasi-miracles make enough of a dissimilarity to outweigh perfect match throughout the future, 
and if I am right that counterfactuals work by similarity, then we can flatly say that if Nixon had 
pressed the button there would have been no quasi-miracle. ([1986b], 61) 



 32

the might-counterfactual is much more plausible than the (L1-wpb) reading. If so, then this is 

indeed a cost of this view. But it is not a devastating cost as one need not flatly reject the 

truth of the might-counterfactual in (L1). 

That is the structure of the response. There is, however, one worry to be addressed. I 

just said that (L1) could be interpreted as  

(L1-wbp) If lightning had struck in R, it would be that: different biological laws are 

possible, 

which is compatible with (L2). (L1-wbp) says that all the most-similar lightning strike worlds 

(call these the L-worlds) are worlds where different biological laws are possible. What does 

this actually mean? Well, the L-worlds are all worlds where the biological laws are the same 

as they are in the actual world. This is what makes (L2) true. But, I claim, the L-worlds are 

also worlds where different biological laws are possible. That is, there are worlds accessible 

to the L-worlds where there are different biological laws. Now, whether or not this is true 

depends on what is meant by ‘accessible’ here. I think we can get the right result if we take 

the worlds accessible relative to the L-worlds to be the set of worlds, L*, that are most 

similar to the actual world under all precisifications of the antecedent of the counterfactual 

we’re considering. So, for instance, we can precisify the antecedent by saying that lightning 

strikes exactly at location x in region R. Then we take all the most-similar worlds relative 

to that precisification and put them in L*. Then precisify the antecedent by saying that 

lightning strikes exactly at location y in region R, and take all the most-similar world relative 

                                                                                                                                                 

But, of course, this seems wrong, and it seems wrong in just the way that (L1) and (L2) seemed wrong. Lewis’s 
response is essentially the one I offered above. 

There is, however, a slight difference. Lewis’s case concerns objective chance, whereas the sort of 
counterexample we are considering does not deal with situations concerning objective chance. This difference, 
however, appears inessential to the kind of response. The heart of the response is that such pairs of ‘would’ and 
‘might’ counterfactuals need not be treated as contradictory. Further, there is something very chance-like going 
on in the purported counterexample. For why do we think that (L1) is true? One might say: ‘If lightning were 
to strike in R, then there’d be some chance that it hits the critical molecule. And if the lightning were to hit the 
critical molecule, then the laws of biology would be different.’ So I think the cases are very analogous. 



 33

to that precisification and put them in L*. The set of worlds in L* are then the worlds 

accessible relative to the L-worlds. Since one way of precisifying the antecedent is a way in 

which the biological laws end up different (when the strike hits the critical event)31 it is true 

to say that relative to the L- worlds, different biological laws are possible. This renders (L1-

wbp) true as desired.32, 
33   

                                                 
31 Notice that the counterfactual, ‘If lightning struck the critical event, then the biological laws would have been 
different,’ comes out true on my account. If lightning hits the critical molecule, then (presumably) the 
fundamental laws entail that biology won’t go as it actually does. This is, after all, what drives the intuition that 
(L1) is true. So, given such a lightning strike, the only way to get biology to go as it actually does is to have 
another fundamental miracle. But any world where this is the case is less similar than the non-biology world in 
virtue of the second fundamental miracle. 
32 A somewhat related worry concerns the following counterfactual, which may seem to come out true on my 
proposed account:  

If it were that A, then there would have been no exceptions to any special science laws.  
This sounds bad, and it isn’t obviously solved by the solution given in the text. What should we say about this? 
The answer is to note that the most-similar A-worlds will be worlds where the fundamental laws are obeyed 
(apart from the small miracle that brings about A). Since we are not assuming that the fundamental laws entail 
the special science laws, there is no reason to think that there will be any most-similar A-worlds where there are 
no exceptions to the special science laws. So we have no reason to think that the counterfactual above will 
come out true. What criterion 3.5 does is to favor worlds where the special science laws hold with some 
exceptions to those in which the special science laws are egregiously violated.    
33 The lightning counterexample considered in the text concerns a very unlikely unlawful world that is ruled out 
of consideration when it shouldn’t be. One might wonder if we don’t get a counterexample going the other 
way, where very likely unlawful worlds are ruled out because of the presence of an unlikely lawful world. That 
is, consider some event, E, that we stipulate will almost certainly result in different biological laws. However, we 
also stipulate, there is one particular way in which E can happen so that the biological laws are the same. Now 
consider the counterfactual:  
 (E1) If E had happened, then the biological laws would have been different. 
This counterfactual might strike one as true. But my account would say that it is false, since the one world 
where the biological laws are the same seems to be ruled as closer to the actual world than the rest. There are 
three main things to say in response to this. First, it is not obvious that (E1) is true. Imagine that I describe a 
way that I’ve soaked a match in water so that it almost certainly won’t light. However, I stipulate that I’ve soaked 
it in such a way that there is still one way in which it could light. Now consider the counterfactual: 
 (M1)  If I had struck this match, then it wouldn’t have lit. 
In much the same way as the counterfactual above, this counterfactual strikes me as false. 

The second thing to say is that E is a very odd sort of event, in that it is stipulated to almost certainly 
result in different biological laws. In virtue of uttering this sort of counterfactual, it is likely that we are moving 
farther away from the standard context of evaluation, which the modified similarity metric is to account for.  

Third, it is important to note that worlds with different biological laws need not transgress criterion 
3.5. Consider a world with no biological properties. The biological laws would have been different in such a 
world, since there would be no such laws. However, such a world does not violate any biological laws, and so is 
not ruled out. Alternatively, consider worlds where there are none of the actual biological properties, but 
different kinds of properties that are similar in certain ways to the actual biological properties. Further, imagine 
that these properties behave in lawlike ways. One might want to call these the biological laws of these worlds. 
These, then, will be worlds that have different biological laws. But these worlds do not violate the actual 
biological laws, and so these worlds will not be ruled out of contention by criterion 3.5. This shows that the 
following counterfactual will come out true, even if we stipulate that there is one way in which E can happen so 
that the biological laws are the same: 
 (E2) If E had happened, then the biological laws might have been different. 



 34

 A second kind of worry concerns what to say in situations where special sciences 

come into conflict.34 For instance, imagine that we have a counterfactual of the form:  

 If it were that A, then it would be that C. 

Further, imagine that there are two A-worlds, b and p tied for similarity on criteria 1-3, and 

that in world b C is true and in world p C is false. Further, b and p differ in the following 

respect: in world b, in addition to the small miracle to bring about A, there is also a violation 

of biological law; in world p, in addition to the small miracle to bring about A, there is a 

violation of psychological law. One might wonder: does the counterfactual come out true or 

false? 

I don’t think there is a unique answer to this question. Sometimes context will make 

clear that one world or the other is the one on which we should be focusing. If we’re at a 

psychology conference, for instance, it is plausible to think that context fixes the 

psychological laws and so world b is preferred. However, if there is no such context to fix 

things in this way, there is a natural way to break ties. Either the biological law or the 

psychological law will have more exceptions in the actual world than the other. If the 

psychological law has more exceptions, then p is closer than b. If the other way, then b is 

closer than p. If they have equal exceptions, then the counterfactual goes indeterminate.    

 This, then, leaves us with the following modified similarity metric: 

(1) It is of the first importance to avoid big, widespread, diverse violations of 

fundamental law. 

(2) It is of the second importance for there to be a spatio-temporal region  

                                                                                                                                                 

This seems to be the correct verdict, in just the same way that the following is correct, when considering the 
match case:  
 (M2) If I had struck this match, then it might not have lit. 
Thanks to an anonymous referee for mentioning this kind of scenario. 
34 Thanks to an anonymous referee for pushing this worry. 



 35

throughout which perfect match of particular fact prevails (with quantitative 

comparisons for worlds with different-sized past match or different-sized future 

match). 

(3) It is of the third importance to avoid small, localized, simple violations of 

fundamental law. 

(3.5) It is of the fourth importance to avoid violation of special science laws (with more exceptionless 

laws preferred in case of ties). 

(4) It is of little or no importance to secure approximate similarity of particular fact. 

8 Conclusion 

This proposal, if successful, blocks Elga’s counterexample. Not only that, but it does so in a 

satisfying way. First, it seems to accord with Lewis’s own views about what a good response 

to the problem would look like: ‘I think the remedy […] is to say that such funny business, 

though not miraculous, makes for dissimilarity in the same way that miracles do.’ (quoted in 

Bennett [2003], p. 296) This solution shows that the funny business is not fundamentally 

miraculous, but that it does make for dissimilarity in the same way that fundamental miracles 

do.  

Second, this solution seems to correctly locate what is odd about Elga’s w3. Given 

Lewis’s original similarity metric, it is true that w3 is just as similar to the actual world as w1. 

But this seems wrong to us. The reason that it seems wrong, I conjecture, is the region of 

anti-entropic behavior. This solution identifies that region as the problem.  

Third, this solution does not simply stipulate the Second Law or the asymmetric 

nature of it. Rather, this solution tells us to take account of any special science laws in the 

evaluation of similarity. Since one true special science law of the actual world is the 

asymmetric Second Law, this fact gets us the asymmetry. 



 36

Fourth, and importantly, the holding of the Second Law at our world can plausibly 

provide the physical basis for the epistemic asymmetry of overdetermination mentioned in 

section 2. At least one true epistemic asymmetry of overdetermination is that there are many 

macroscopic approximate determinates after an event (the smoking gun, the bloody glove, 

the footprint, etc.) and very few (if any) macroscopic approximate determinates before an 

event. A physical world that has such an epistemic asymmetry of overdetermination will 

have it in virtue of some physical asymmetric macroscopic regularities. These asymmetric 

macroscopic regularities are plausibly the regularities that special science laws in general, and 

the Second Law in particular, capture. Thus, worlds that exhibit small-miracle convergence 

to the actual world do lack certain macroscopic asymmetric regularities of the actual world. 

And a similarity metric that takes into account the Second Law is able to give a principled 

reason for ruling out such easy convergence worlds. 

 

Acknowledgements 

Special thanks to Phil Bricker and Chris Meacham for very helpful discussion on issues 

related to this paper. Thanks also to the participants of the 2008 Central APA session where 

a version of this paper was read, especially Valia Allori who provided written comments. I 

also appreciate the helpful comments from the participants of the UMass Philosophy 

Graduate Student Workshop, and two anonymous referees for this journal. 

 

References 
Albert, D. Z. [2000]: Time & Chance, Cambridge, MA: Harvard University Press. 
Albert, D. Z. [1994]: ‘The Foundations of Quantum Mechanics and the Approach to  

Thermodynamic Equilibrium’, British Journal for the Philosophy of Science, 45,  
pp. 669-677. 

Bennett, J. [2003]: A Philosophical Guide to Conditionals, New York: Oxford  
University Press.  

Bennett, J. [1984]: ‘Counterfactuals and Temporal Direction’, Philosophical  



 37

Review, 93, pp. 57-92. 
Buckingham, A. D. [1964]: The Laws and Applications of Thermodynamics, Oxford:  

Pergamon Press. 
Burshtein, A. I. [1996]: Introduction to Thermodynamics and Kinetic Theory of Matter,  

New York: John Wiley & Sons, Inc. 
Callender, C. [2006]: ‘Thermodynamic Asymmetry in Time’, The Stanford  

Encyclopedia of Philosophy (Fall 2006 Edition), Edward N. Zalta (ed.), URL =  
<http://plato.stanford.edu/archives/fall2006/entries/time-thermo/>. 

Callender, C. [2001]: ‘Taking Thermodynamics Too Seriously’, Studies in the History  
and Philosophy of Science Part B, 32, pp. 539-553. 

Callender, C. [1997]: ‘What is ‘The Problem of the Direction of Time’?’, Philosophy  
of Science, 64, pp. S223-S234. 

Craig, N. C. [1992]: Entropy Analysis: An Introduction to Chemical Thermodynamics, 
New York: Wiley-VCH. 

Earman, J. [1986]: A Primer on Determinism, Dordrecht: D. Reidel Publishing Company. 
Elga, A. [2000]: ‘Statistical Mechanics and the Asymmetry of Counterfactual  

Dependence’, Philosophy of Science, 68, pp. S313-S324. 
Fodor, J. [1974]: ‘Special Sciences’, Synthese, 28, pp. 97-112. 
Gokcen, N. A. & Reddy, R. G. [1996]: Thermodynamics (Second Edition). New York:  

Plenum Press. 
Greiner, W. et. al. [1995]: Thermodynamics and Statistical Mechanics, Translated: Dirk  

Rischke, New York: Springer-Verlag. 
Hawthorne, J. [2007]: ‘Craziness and Metasemantics’, Philosophical Review, 116, pp.  

427-440. 
Lange, M. [2000a]: Natural Laws in Scientific Practice, New York: Oxford University  

Press. 
Lange, M. [2000b]: ‘Salience, Supervenience, and Layer Cakes in Sellars’s Scientific  

Realism, McDowell’s Moral Realism, and the Philosophy of Mind’, Philosophical 
Studies, 101, pp. 213-251. 

Lewis, D. [1999]: Papers in Metaphysics and Epistemology, New York: Cambridge  
University Press.  

Lewis, D. [1986a]: On the Plurality of Worlds, Oxford: Blackwell. 
Lewis, D. [1986b]: Philosophical Papers, vol. II, New York: Oxford University Press. 
Lewis, D. [1973]: Counterfactuals, Malden, MA: Blackwell. 
Lieb, E. H. & Yngvason, J. [2000]: ‘A Fresh Look at Entropy and the Second Law of  

Thermodynamics’, Physics Today, 53, pp. 32-37. 
Loewer, B. [forthcoming]: ‘Why There is Anything Except Physics’, in Being Reduced:  

New Essays on Reduction, Explanation, and Causation, Hohwy, J. and Kallestrup, J. (eds), 
New York: Oxford University Press. 

Loewer, B. [2007]: ‘Counterfactuals and the Second Law’, in Causation, Physics,  
and the Constitution of Reality: Russell's Republic Revisited, Price, H. and Corry, R. (eds), 
New York: Oxford University Press, pp. 293-326. 

Loewer, B. [2001]:  ‘Determinism and Chance’, Studies in the History and Philosophy of  
Modern Physics, 32, pp. 609-620. 

Sanford, D. H. [1989]: If P, then Q: conditionals and the foundations of reasoning,  
London: Routledge. 

Schaffer, J. [2007]: ‘Deterministic Chance’, British Journal for the Philosophy of  
Science, 58, pp. 113-140. 



 38

Schaffer, J. [2004]: ‘Two Conceptions of Sparse Properties’, Pacific Philosophical  
Quarterly, 85, pp. 92-102.