Deterministic chance


Deterministic Chance 
Luke Glynn 

ABSTRACT 

I argue that there are non-trivial objective chances (that is, objective chances other than 0 
and I) even in deterministic worlds. The argument is straightforward. I observe that there 
are probabilistic special scientific laws even in deterministic worlds. These laws project 
non-trivial probabilities for the events that they concern. And these probabilities play the 
chance role and so shou ld be regarded as chances as opposed, for example, to epistemic 
probabilities or credences. The supposition of non-trivial deterministic chances might 
seem to land us in contradiction . The fundamental laws of deterministic worlds project 
trivial probabilities for the very same events that are assigned non-trivial probabilities 
by the special scientific laws. I argue that any appearance of tension is dissolved by 
recognition of the level-relativity of chances. There is therefore no obstacle to accepting 
non-trivial chance-role-playing deterministic probabilities as genuine chances. 

1 Introduction 

2 Schaffer's Incompatibilist Argument 

2.1 Chance and credence 

2.2 Chance and possibility 

2.3 Chance and la ws 

3 Special Scientific Laws 

3.1 Probabilistic special scientific laws in deterministic worlds 

3.2 Lewis s Humean analysis of laws 
3.3 Special scientific laws and the law role 

4 Deterministic Chance 

4.1 Chance and laws again 

4.2 Chance and credence again 

4.3 Chance and possibility again 

5 Chance and Causation 

6 Conclusion 

Appendix: Times, Levels, and Chance Setups 

http://bjps.oxfordjournals.org/
http://nbn-resolving.de/urn:nbn:de:bsz:352-140984


52 

1 Introduction 

It has often been assumed without argument that determinism is incompatible 
with non-trivial objective chance. Popper ([1982], p. 105) , for example, simply 
says: 

[O]bjective physical probabilities a re incompatible with determinism ; [ .. . ] 

In a similar vein Lewis ([ 1986c], p. 118) states that: 

if the chance [of a fair coin landing heads] is zero or one, [ . .. ] then it 
cannot also be 50%. To the question how chance can be reconciled with 
determinism, or to the question how disparate chances can be reconciled 
with one a nother, my answer is : it can '{ be done. 

Some argument for such claims is needed . Deterministic physical theories 
such as classical statistical mechanics (CSM) yield non-trivial probabilities that 
are used by physicists in prediction and explanation. Both the manner of their 
genesis and the nature of their employment make these probabilities good 
candidates to be considered obj ective chances. I 

But Schaffer ([2007]; all subsequent citations to Schaffer will refer to this 
paper) has recently argued that they are not chances. His argument is the most 
sophisticated and well developed such argument currently to be found in the 

literature. He argues that although deterministic probabilities such as those 
generated by CSM may be formally eligible to count as chances-they are 
generated by a function from propositions, times, and worlds onto the closed 
unit interval in accordance with the axioms of the probability calculus- they 

do not qualify as chances because they fail to play the chance role. Generating 
values that play the chance role is what distinguishes the chance function from 
th e many other formally eligible function s that no-one would regard as the 
chance fun ct ion (Schaffer, p. 123). 

Schaffer claims that the role of chance is characterized by its connections 

with 'credence, possibility, futurity, intrinsicness, la whood, and causation' . He 
argues tha t, in a deterministic world, a chance function outputting non-trivial 
values would violate three of these connections: namely, those from chance to 
credence, p ossibility a nd lawhood (p. 132).2 By contrast, a function th at outputs 
just trivial valu es in deterministic worlds plays the chance role perfectly, and so 
this should be regarded as the chance function (p. 127). Schaffer's conclusion 

is that th ere are therefore no non -trivial deterministic chances. Hi s argument 
will be outlined in Section 2. 

1 This point is emph asized by Loewer ([2001], see esp. p. 610). 
2 My concern in thi s paper is with what Schaffer (p. 120) ca ll s poslerior determini s ti c chances: 

chances co ncernin g eve nt s occurring afier the fir st mome nt o f th e universe. Schaffer's case aga in st 
il/ilia/ deterministic cha nces- conce rning eve nt s occurring al th e fir st moment of a deterministic 
universe- is somewhat different , a nd I won 't defend such c hances here. 



53 

In Sections 3-4 I shall seek to show that Schaffer is wrong: there are 
non-trivial deterministic probability functions that underwrite the connections 
from chance to credence, possibility and lawhood, and that therefore play the 
chance role. Important to the demonstration that this is so is the demonstra-
tion that chance is level-relative (this notion of level-relativity is explicated in 
Section 4). Recognition of the level-relativity of chances allows us to see the 
flaws in Schaffer's arguments. 3 

Not only are there non-trivial deterministic probability functions that play 
the chance role, some such functions play it better than any trivial deterministic 
probability function. In Sections 4.1 and 4.2 I argue that, for at least some deter-
ministic worlds, only a function that outputs non-trivial values can underwrite 
the connections from chance to lawhood and credence. 

I seek to reinforce the case for non-trivial objective chances in Section 5 by 
considering one of the remaining three connections discussed by Schaffer: that 
from chance to causation . Schaffer says (p. 132) he sees 'no problem' regarding 
the consistency of a non-trivial deterministic chance function with this con-
nection, nor with the other two-from chance to futurity and intrinsicness. I 
argue that, not only is Schaffer correct in this, but that there are good rea-
sons to think that a trivial deterministic chance function is inconsistent with 
the chance--causation connection. The conclusion (Section 6) is that there are 
non-trivial deterministic chances. 

2 Schaffer's Incompatibilist Argument 

I will now review Schaffer's argument that (non-trivial) deterministic chances 
would violate the connections from chance to credence, possibility and lawhood 
and that chance is therefore incompatible with determinism. 

J In a footnote, SchatTer (p. 115n.) states that he is assuming chance to be a function of only 
three arguments : propositions, worlds, and times. He says (ibid.) that 'whether determinism is 
compatible with a "chance" function that is relativized to [ ... ] further inputs should be considered 
a separate question not addressed in the main text. ' 

Perhaps SchatTer is correct that, on the assumption that the chance jilllClion has only three 
(//gllll1ents , it follows that there are no deterministic chances. But what I am concerned with 
is whether there are deterministic chances, not whether there are deterministic chances on this 
assumption. Consequently, I regard evidence that there is a fourth argument (such as a level) as 
relevant to my topic. Such evidence will be presented in the course of this paper. 

SchatTer suggests that he has some justification for making the three·argument assumption. 
He says (ibid.) that in his paper the assumption 'will be defended indirectly [ .. . ] insofar as the 
role of chance [ . .. ] will prove explicable in ways that require no further inputs. ' But I shall argue 
below that SchatTer's explication of the role of chance is flawed, and that the flaws can only be 
remedied by explicitly relativizing chance to a fourth input. 

I am far from being the first to argue that chance is relative to more than just propositions, 
times, and worlds. Hajek ([2003a], [2003b], [2007]), for example, has argued vigorously (and in 
my view convincingly) that chance must be relativized to a chance setup or reference class. In the 
Appendix , I discuss the relation between my claim that chance is level·relative, and Hiljek's that 
chance is setup-relative. 



54 

2.1 Chance and credence 

Schaffer takes the chance--credence connection to be adequately captured by 
Lewis's ([1986b]) Principal Principle (PP), which can be stated as follows (ibid., 
p. 87). Let C be any reasonable initial credence function, let t be any time and let 
w be any world. Let Ch tw be the chance function (which receives time and world 
indices because of the time- and world-relativity of chance4), let x be any real 
number in the closed unit interval and let p be any proposition in the domain 
of the chance function. Finally, let Xbe the proposition that Chtw(p) = x, and 
let E be any proposition compatible with Xthat is admissible at t and w. Then: 

CCp I X. E) = x (PP) 

The idea is that if one started with a reasonable initial credence function (and 
updated by conditioning) and one learned that Chtw(p) = x and if the rest of 
one's evidence E were admissible at time t (and consistent with the proposition 
that Chtw(p) = x), then one would have credence in p equal to x. 

To get a handle on the content of the PP, we need to know what sort of 
evidence counts as 'admissible'. Lewis's characterization of admissible infor-
mation is as follows: 

Admissible propositions are the sort of information whose impact on cre-
dence about outcomes comes entirely by way of credence about the chances 
of those outcomes. (ibid., p. 92) 

Lewis suggests (ibid., pp. 92- 6) that two different sorts of information are gen-
erally admissible at a time t and world w: historical information (or information 
about matters of particular fact at times no later than I) and information about 
the laws of w . This allows him to reformulate the P P in a manner that 'will 
prove easier to use' (ibid., p. 98), as follows. 

Let Htw be a proposition giving the complete history of w up to t and let Lw 
be a proposition giving the laws of nature that obtain at w. These propositions 
are both admissible at t and wand so, Lewis claims (ibid., p. 96), is their 
conjunction. Suppose that X is true: that is, Chtw(p) = x. Then, since both 
Hrw .Lw and X hold at w, they are compatible. The conjunction Hrw.L w can 
therefore be substituted for the proposition E to yield (ibid., p. 97): 

(PP') 

Indeed, according to Lewis (ibid., p. 97, [1994], pp. 477- 78), the laws of w 
together with the initial history of w through t entail the chances that obtain at 
t and w. Hence, Lw.Htw entails X. So X.Lw.Htw can be simplified to L w. Htw . 

4 Lewi s (ibid., pp. 9 1- 2) advances arguments for thi s time- and world-relativity. The issue of whether 
chance isjilll!iamell/ally time-relative will be taken up in the Appendix. 



55 

Since it is also true that Chtw(p) = x, PPcan be reformulated as follows (Lewis 
[1986b], pp. 96-7, [1994] , p. 487): 

(RPP) 

The idea is that the laws of w together with the initial history of w through 
t entail the chances obtaining at wand t and that reasonable credence for 
someone whose evidence includes just the laws and initial history (and who 
updates by conditioning on her evidence) is equal to the chance of p that they 
entail. 

We are now in a position to consider Schaffer's argument for the incompati-
bility of non-trivial deterministic chances with the connection from chance to 
rational credence. His argument is as follows ([2007], p. 128; cf. Hoefer [2007], 
pp.558- 9). 

Suppose that Pe is the proposition that some event e occurs,s and that w is a 
deterministic world . Suppose, moreover, that Chtw(Pe) is a non -trivia l chance: 
that is I > Chtw(Pe) > O. 

Now either e occurs in w or it does not. Suppose it does. Then, since w is 
deterministic, the laws of w together with the history of w up to (and includ-
ing) any time t entail Pe. Accordingly, for any reasonable credence function , 

C(PeILw.Htw) = I. It follows immediately by the RPP that Chtw(Pe) = I. 
Suppose, on the other hand, that e does not occur in w. Then the laws of w 

together with its history through t entail ~Pe. Accordingly, for any reasonable 
credence function, C(PeILw . lftw) = O. It follows immediately by the RPP that 
Chtw(Pe) = O. 

So, by the assumption that Chtw(Pe) is a non-trivial deterministic chance, 
we have that I > Chtw(Pe) > O. But, by the RPP, we have that Chtw(Pe) equals 
either I or 0 (depending on whether or not e occurs in w). Contradiction ! 
Schaffer's conclusion: non-trivial deterministic chance assignments are incom-
patible with the connection from chance to rational credence captured by the 
PP (of which the RPP is a reformulation). 

2.2 Chance and possibility 

Schaffer argues that non-trivial deterministic chances would a lso violate the 
connection from chance to possibility. He claims (p. 124) that this connection 
is captured by what he ca lls the Realization Principle (RP):6 

5 Schaffer ([2007] , p. 121 n) exp licit ly restricts hi s attention to determini stic chances pertaining to 
propositions concerning individual , mom entary event occur rences. I sha ll follow him in thi s. 

6 Th e RP is a strengthened version of Bi ge low, Co ll ins, and Pargetter's Basic Chali ce Principle 
([ 1993], p. 459). 



56 

(RP) If Chtw(p) > 0, then there exists a world w' such that (i) w' matches 
w in laws; (ii) w' matches w in occurrent history up until time I; (iii) p is 
true at w' . 

Schaffer ([2007], p. 130) argues that non-trivial deterministic chances are in-
consistent with the RP as follows. Suppose that w is a deterministic world and 
Ch,w(Pe) is a non-trivial chance: that is I > Ch,w(Pe) > O. 

Now either e occurs in w or it does not.lfit does then, since w is deterministic, 
the laws and initial history of w entail P .. so there cannot be a world that has 
the same laws and initial history as w at which ~ Pe is true. Therefore, by the RP, 
Ch,w( ~ Pe) = 0 from which it follows by complementation that Ch,w(Pe) = I. 

If, on the other hand, e does not occur at w, then the laws and initial history 
of w entail ~ Pc> so there cannot be a world with the same laws and initial 
history as w at which Pe is true. It follows directly by the RP that Ch,w(Pe) = O. 

So, by the assumption that Ch,w(Pe) is a non-trivial deterministic chance, 
we have that I > Ch,w(Pe) > O. But, by the RP, we have that Ch,w(Pe) equals 
either I or 0 (depending on whether e occurs in w). Contradiction! Schaffer's 
conclusion: non-trivial deterministic chance assignments are incompatible with 
the connection between chance and possibility captured by the RP. 

2.3 Chance and laws 

Finally, Schaffer argues that non-trivial deterministic chances are incompatible 
with the connection from chance to lawhood. He claims (p. 126) that this 
connection is captured by the lawful magnitude principle (LMP): 

(LMP) If Chtw(p) = x, then the conjunction of fitw (a proposition giving 
the history of w up until time t) with the laws of w entails that Chtw(p) = x. 

The LMP says, in effect, that chances are lawfully projected magnitudes. Schaf-
fer argues (p. 130) that non-trivial deterministic chances would be inconsistent 
with the LMP, since 'laws at deterministic worlds do not project chances'. 

The idea is as follows. Suppose that Ch,w(Pe) is a non-trivial determinis-
tic chance: that is I > Ch,w(Pe) > O. Since w is deterministic, Lw.H,w entai ls 
either Ch,w(Pe) = I or Ch,w(Pe) = 0 depending on whether e occurs at w. 
It follows that the only chance assignments compatible with the LMP are 

Ch,w(Pe) = I or Ch,w(Pe) = O. But now we have that I > Ch,w(Pe) > 0 and 
that either Ch,w(Pe) = I or Ch,w(Pe) = O. Contradiction! Schaffer's conclu-
sion: non-trivial deterministic chance assignments are incompatible with the 
connection between chance and laws captured by the LMP. 

Since a (non-trivial) deterministic chance function would 'sever the con-
nections from chance to credence, possibility, and lawhood ' (p. 132), Schaffer 
concludes that ' [t]his is hardly a viable conception of chance' (ibid.). 



57 

In Sections 3 and 4, I argue that Schaffer is wrong: there are non-trivial 
deterministic chance functions that are consistent with all three of these con-
nections. Recognition of the level-relativity of chance allows us to see that this is 
so and undermines Schaffer's apparent demonstration that these connections, 
when taken together with the supposition of non-trivial deterministic chances, 
land us in contradiction . 

Indeed I will argue that, once the level-relativity of chance is recognized, 
it can be seen that only a chance function that outputs non-trivial values in 
deterministic worlds can underwrite the connections from chance to credence 
and lawhood. 

3 Special Scientific Laws 

The main reason to think that there are non-trivial objective chances even in 
fundamentally deterministic worlds is that there exist probabilistic high-level 
or special scientific laws even in such worlds (Section 3.1).1 The probabili-
ties projected by these laws should be regarded as genuine, objective chances 
because the laws in question are genuine, objective laws. Not only are they 
accommodated as such by Lewis's Humean analysis of lawhood (Section 3.2) 
but, because they play the law role, they must similarly be accommodated by 
any adequate account of lawhood (Section 3.3). 

3.1 Probabilistic special scientific laws in deterministic worlds 

Quantum mechanics- at least on standard, 'collapse' interpretations (e.g., 
Copenhagen and GRW)-indicates that the fundamental dynamics of our 
world is probabilistic. But it is not just fundamental physics that is proba-
bilistic. Many of the high-level or special sciences also give probabilistic laws 
for events falling under their purview. As already noted, statistical mechanics 
gives such laws. So does Mendelian genetics. And probabilistic functional laws 
are encoded in the models of economists and meteorologists. Together with 
the initial history of the world, these special scientific laws entail non-trivial 
chances for the events that they concern. 

The Mendelian genetic laws of Segregation and of Independent Assort-
ment, for example, give a chance 0.25 for a dihybrid cross between two parents 

7 Or at least in some such worlds. There might be deterministic worlds lacking probabilistic special 
sci en tiflc laws. I do not argue that the re are non-trivial chances in these worlds. My thesis is 
merely that there are some deterministic worlds with non-trivial chances. In particular, there are 
non-trivial chances in those deterministic worlds that ours might have turned out to be (if it 
had turned out to be Newtonian) or might still turn out to be (if it turns out to be Bohmian); 
worlds that are deterministic at the microphysical level but at which there are probabi listic specia l 
scientific laws. 



58 

heterozygous for each (binary) trait yielding a child that is homozygous for 
each trait. So suppose that Jim and Jill are common garden pea plants (Pisum 
sativum) heterozygous for both pea shape and colour (both binary variables, 
with round and yellow the dominant alleles, and wrinkled and green the recessive 
alleles), and Tom is a plant produced by crossing Jim and Jill. Let t be the time 
of crossing, and p the proposition that Tom is homozygous for both shape and 
colour. Then the Mendelian laws together with the history of the world through 
t (which includes the fact that Jim and Jill are heterozygous for both shape and 
colour and the fact that they are crossed) entail a chance 0.25 for Tom's being 
homozygous for both pea shape and colour. 

In most cases, the special sciences make no presupposition about whether 
the fundamental dynamics of the world is deterministic or indeterministic. 
Where they do, as in CSM, the assumption is often one of microphysical 
determinism . It follows from the compatibility of the probabilistic special 
sciences with fundamental determinism that there exist fundamentally deter-
ministic worlds (Newtonian or Bohmian worlds, perhaps) with probabilistic 
special scientific laws. 

A lot of work remains to be done to show that the non-trivial probabili-
ties projected by the probabilistic special scientific laws of these deterministic 
worlds are genuine objective chances. The opponent of deterministic chance 
will presumably seek to dismiss this claim, arguing that such probabilities are 
merely epistemic (indeed Schaffer makes precisely this argument, pp. 136- 9). 

The key to showing that these probabilities are objective chances is, of course, 
to show that they play the objective chance role. This will at least involve defus-
ing Schaffer's arguments, reviewed in the previous section, that no non-trivial 
deterministic probability function can play the chance role. I shall argue that 
there are some such functions that can and that, moreover, trivial deterministic 
functions cannot. 

It seems that the best place to start in showing that non-trivial deterministic 
probability functions can play the chance role will be with Schaffer's claim 
(considered in Section 2.3 above) that such functions violate the chance- law 
connection because the laws of deterministic worlds don't project chances. Af-
ter all, it has just been argued that Schaffer is simply wrong in this claim. True, 
the fundamental laws of deterministic worlds don't project chances. But not 
all laws are fundamental. And it has just been argued that there are proba-
bilistic high-level, special scientific laws even in fundamentally deterministic 
worlds. 

Schaffer is aware of the vulnerability of his position to this kind of objection. 
His reaction (pp. 130- 2) is to argue that , on Lewis's Humean view of laws, 
there are good reasons for regarding these probabilistic special scientific ' laws ' 
as failing to be genuine laws. Before considering his argument, it is worth getting 
a bit clearer on Lewis's analysis. 



59 

3.2 Lewis's Humean analysis of laws 

Lewis's analysis oflaws ([1994], pp. 478,480) is as follows. Consider all deductive 
systems whose theorems pertain to what happens in history, in the sense that 
they give either the outcomes or chances of outcomes in various situations. 
Exclude those systems whose theorems aren't true in what they say about 
outcomes. (Lewis is attempting to deliver a simultaneous analysis of laws and 
chances, so it is not yet required-that the theorems must be true in what they 
say about the chances.) Also, exclude any that say what an outcome will be 
without also saying that the outcome never had a chance of not coming about. 

The remaining systems may differ in simplicity, strength (or informativeness), 
and fit (or the chance that they assign to the actual course of history). And there 
will be trade-offs between these virtues: simpler systems may be less strong or 
fit less well, and so on. According to Lewis: 

The best system is the system that gets the best balance of all three. [ ... J 
[T]he laws are those regularities that are theorems of the best system [. - .] 
[and] the chances are what the probabilistic laws of the best system say they 
are. (ibid., p. 480) 

If it can be shown that (at least some of) the special scientific laws are theorems 
of the Best System, then it will follow (pace Schaffer) that Lewis's Humean 
analysis accommodates these special scientific laws as genuine, objective laws. 

Indeed, there are good reasons to think that (at least some) special scientific 
laws are theorems of the Best System. In particular, it seems that a system that 
yields the special scientific laws as theorems will be much stronger or more 
informative than one that yields merely the fundamental, microphysical laws. 
This is because the microphysical laws tend to fall silent about the higher-level 
properties that the special scientific laws relate. 

The high-level properties in question are typically multiply realisable at the 
microphysical level. That is, different distributions of microphysical properties 
may realize the same high-level property. So, for example, two individuals a 
and b sharing some biological property F could be in rather different micro-
physical states, consisting of differing distributions of particles with differing 
values of mass, spin, charge, etc. If the property F were to be characterized in 
microphysical vocabulary, it would be in terms of a very long disjunction with 
each disjunct- itself a highly complex conjunction of propositions attributing 
microphysically simple properties to particles-corresponding to one possible 
microphysical realization of F (cf. Hoefer [2007], p. 593). 

This relationship of realization grounds the distinction between higher and 
lower levels_ Two or more sciences can be regarded as characterizing distinct 
levels when various distributions of the properties of concern to one realize 
those of concern to the other. Where this is not the case, the sciences in question 
should be regarded as each providing partial characterizations of the same level. 



60 

Each level, consisting of the distribution of a certain set of properties together 
with the nomic relations between those properties, constitutes a relatively closed 
system in the sense that adequate (non-reductive) explanations for the instanti -
ation of any property within that set can be given in terms of the instantiation 
of other properties in that set together with the nomic connections between 
them. 

Where two or more sciences partially characterize the same level, each taken 
individually will lack explanatory closedness. Climate change, for instance, can-
not be adequately explained without appeal to human industrial activity as well 
as geological a nd cosmological factors. This reflects the fact that climatology, 
economic sociology, geology, and cosmology do not each characterize distinct 
levels, but rather each provides a partial characterization of a certain relatively 
'high' level. 

The multiple realizability of higher-level properties at the microphysical level 
is the key to understanding how special scientific laws enhance the informative-
ness of a system that entails them. Suppose that the biological property F can 
be realized by anyone of the complex microphysical states a, {3, y, etc. And 
suppose that there is some other biological property, G, that can be realized by 
anyone of the complex microphysical states a', {3', y', etc. For each particular 
microphysical realization of F, the microphysical laws will perhaps give some 
well-defined chance of the system's going on to exhibit one of the microphysical 
states that realizes G. However, it cannot be expected that the microphysical 
laws alone will tell us anything about the relationship between F and G in 
general. Specifically, it cannot be expected that they will tell us the chance that 
G is instantiated given that F is insta ntiated, since they themselves do not give 
us a probability distribution over the states a, {3, y, etc., conditional upon F's 
being instantiated. 

Consequently, a system that entails only the microphysical laws will lack a 
certain amount of strength or informativeness in Lewis's sense: it will fail to say 
'either what will happen or what the chances will be when situations of a certain 
kind arise' ([ 1994], p. 480). The kinds of situation include, of course, situations 
of kind F. This would be a heavy cost indeed where F and G correspond to 
important biological kinds as is the case, for example, if F is the property of 
being a crossing of two parents heterozygous for a given trait, and G is the 
property of the offspring's being homozygous for that trait. 

The fact that the microphysical laws fall silent here means that there is a gap 
to be filled by the special scientific laws. In this case the Mendelian laws fill 
the gap. The addition of axioms that entail these laws will augment a system's 
strength because the resulting system will tell us what the chance of G is in 
situations of kind F: namely (on the current interpretation of F and G) 0.5. 

Nor need the addition, to a system, of axioms required to entail the special 
scientific laws cost much in terms of simplicity. What we need to add to the 



61 

axioms, in order to get the special scientific laws to fall out as theorems, is a 
function that takes macrostates as inputs and yields probability distributions 
over regions of microphysical phase space as outputs (this would yield the 
requisite probability distribution over the microphysical states a, {J, y, etc., 
conditional upon F's being instantiated , from which an overall chance of G 
given F could be derived). 

A function that yields a uniform probability distribution over those regions 
compossible with the macrostate might result in the best fit with the frequencies 
observed at a world like ours. If, for example, the various possible microphys-
ical realizations of F lead to microphysical realizations of G with an average 
frequency of 0.5 then a uniform probability distribution over microphysical 
realizations of F will yield a chance of G given F close to the actual frequency 
of Gs amongst Fs. In any case, the function could be tweaked to ensure the 
best fit with the observed frequencies in our world. 

The simple addition of such a function to the axioms would thus increase 
the strength of the system greatly by ensuring that the system tells us what the 
chances of G will be when situations of kind F arise and, in general, ensuring 
that it tells us what the chances will be when situations of kind X arise, where X 
is any microphysically disjunctive high-level kind. It consequently seems likely 
that such a system will come out best. 

The illustration just given of how the Lewisian Best System Analysis can 
accommodate special scientific laws is similar to Loewer's attempt to show 
that the Best System for a Newtonian world will entail CSM. Loewer ([2001]) 
considers a formulation ofCSM given by Albert ([2001]). Albert's formulation 
consists of three postulates : (i) the Newtonian dynamical laws, (ii) a uniform 
probability distribution over the possible points in microphysical phase space at 
the beginning of the universe, and (iii) a statement characterizing the beginning 
of the universe as a low-entropy state. I shall borrow the terminology of Schaffer 
([2007], p. 122) in calling postulates (2) and (3) respectively the 'Statistical 
Postulate' and the 'Past Hypothesis'. 

Albert shows that from these three postulates the whole of statistical mechan-
ics follows, as well as probabilistic versions of the principles of thermodynamics. 
Thus Loewer ([2001], p. 618) says: 

[TJhis package is a putative Best System [ . . . j. The contingent generalisa-
tions it entails a re laws and the chance statements it entails give the chances. 
It is simple and it is enormously informative. [ ... J By being part of the Best 
System the probability distribution earns its status as a law and is thus 
able to confer lawfulness on those generalisations that it (together with the 
dynamical laws) entails. 

The Albert package is similar to the system sketched above in its combina-
tion of the fundamenta l dynamical laws with a statistical postulate specifying 



62 

a uniform probability distribution over points in microphysical phase-space. 
The main differences are, first, that whilst the Albert package just gives a single 
probabi lity distribution over phase-space points at the beginning of the uni-
verse, the postulate I described is an atemporal function from macrostates to 
probability distributions. Second, the Albert package includes the low-entropy 
condition.8 

Schaffer ([2007], pp. 130- 1) advances an argument against the view that 
the Best System will entail the special scientific laws. He argues that, even 
though the addition of some statistical postulate to a system may result in a 
gain in informativeness vis-a-vis one that entails the microphysical laws alone, 
these two systems aren't the only competitors. He suggests that we consider 
an alternative package of the fundamental dynamical laws together with the 
Precise Initial Conditions. Where the microphysical laws are deterministic, this 
package will have great strength, since it 'entails every single detail of the entire 
history of the world' (p. 131). 

A crucial question here concerns the simplicity of this latter system vis-a-vis 
one having just the fundamental dynamic laws and a statistical postulate as 
axioms. Hoefer ([2007], p. 560) puts the question well when he asks, regarding 
the Precise Initial Conditions, 

8 This condition creales a difficulty for lhe view that the Albert package is the Best System for 
some not -too -distant Newtonian world. The problem is that, in an earlier paper, Lewis restricts 
candidates for best syslemhood to those systems whose axioms refer only to perfectly natural 
properties (Lewis [1983], pp. 368- 9). Lewis hold s thaI the perfectly natural properties are those 
to be discovered by fundamental physics (ibid. , pp. 365, 368) . And , as Schaffer ([2007], p. 130) 
points oul: 

The Albert package contains predicates such as ' low entropy' that refer to properties 
that are not perfectly natural - in microphysical vocabulary, that property is infinitely 
disjunctive. Hence the Albert package is not even ill the runlling for the Lewis laws. It 
is in eli gible from the start. 

I think thaI Lewis 's restriction is unduly severe (he does not mention it in his final statement 
of his Best System Analysis [1994]). His ([ 1983], p. 367)justification for it was th aI the simp li city 
of a system is re lative to the vocabulary in which it is expressed and that, by employing a very 
unnatural predicate, we might make a strong system very (syntactically) simple indeed. The 
concern was that, if just allY predicates are a ll owed , it is difficult to see how simplicity is a virlue 
of a system , and how it can put any constraints upon a system's acceptabi lity. 

But il does not follow that a restriction to perfectly natural predicates is required. Naturalness, 
as Lewis recognizes (ibid., p. 368), admits of degrees. And the fact that, as Lewis also recognizes 
(ibid.), our language 'has words mostly for not -loo-unnatural properties' suggests that we are 
fairly adept at distinguishing reasonably natural properties from unnatural ones. Simp li city will 
not be a vacuou s c riterion so long as we require that the predica tes employed in the axioms of th e 
candidate systems refer to reasonably natural prope rties. Requiring reasollable natura lnes s may 
int roduce a certain amount of indeterminacy or subjectivit y into the Best System Analysis but , 
as Lewis ([1994], p. 479) is aware, there is already indeterminacy and subjectivity in th e analysis 
as a res ult of its employment of the notion s of simplicity, strength, and balance. 

Rela xing a little the requirement of perfect naturalness of the predicates employed in the axioms 
is perhaps reasonabl e if the consequence of maintaining it in its strictest form is the exclusion of 
certain systems, such as the Albert package, which yield great gains in strength . 



(0)0 they increase the complexity of the system infinitely, or by just one 
'proposition', or some amount in between? Lewis's explication does not 
answer these questions, and intuition does not seem to supply a ready 
answer either. 

63 

It is clear that different explications of the notion of simplicity may have strong 
implications for which system comes out as best. 

But, in any case, Schaffer is wrong to claim that the package of the micro-
physical laws together with the Precise Initial Conditions is 'maximally strong' 
([2007], p. 132). The package is not maximally strong on Lewis's explication 
of the relevant notion of strength. For, as has already been seen, it fails to 
say 'either what will happen or what the chances will be when situations of a 
certain kind arise'. For example, it fails to tell us what will happen or what the 
chances are when situations of the microphysically disjunctive biological kind 
F arise, as opposed to situations of the realizing microphysical kinds lX, {3, y, 
etc. 

3.3 Special scientific laws and the law role 

Following Schaffer and Loewer, I have focused on the Lewisian approach to 
laws, but there seems no good reason why other Humean approaches, as well 
as non-Humean approaches to laws, should not also be able to accommodate 
special scientific laws as genuine. Indeed there is good reason to think that, in 
order to be fully adequate, any analysis of laws must be able to accommodate 
special scientific laws. The reason is that these laws seem to play the law role just 
as well as the microphysical laws. In particular, the special scientific laws fulfil 
all the usual criteria for genuine lawhood by supporting counterfactuals, being 
confirmed by their instances and underwriting explanations and predictions. 

For example, it is true in virtue of the Mendelian laws that if I had conducted 
a dihybrid cross of two pea plants heterozygous for two binary traits, then there 
would have been a chance 0.25 of the offspring being homozygous for each 
trait. Moreover, if I know the Mendelian laws, then I should have confidence 
of degree 0.25 in the prediction that a given pea plant resulting from such a 
cross will be homozygous for each trait. Furthermore, if a resulting pea plant 
does in fact turn out to be homozygous for each trait, then a good explanation 
(if there is such a thing as a good covering law explanation) would be that I 
conducted the cross and that the Mendelian laws indicate that there was a 0.25 
chance that this outcome would result. 

Finally, the Mendelian laws derive inductive confirmation from experimental 
results in just the same manner as do microphysical laws. Mendel ([1866]) 
himself conducted experimental crossings of29,000 pea plants. His experiments 
with mixing one trait with another consistently yielded a 3: I ratio between 
dominant and recessive phenotypes, whilst his experiments with mixing two 



64 

traits resulted in 9:3:3: I ratios. These experiments are rightly seen as having 
conferred rather good inductive confirmation upon the laws named after him.9 

Someone might object to the foregoing along the following lines: 'If the 
microphysical laws project different chances from the Mendelian laws for the 
outcomes of dihybrid crosses, then the counterfactual "if I had conducted 
a cross, there would have been a chance 0.25" is false and, moreover, one 
should not (if one knew the microphysical laws) have confidence 0.25 in the 
outcome, nor does the 0.25 chance really explain the outcome'. The objector 
might instead maintain that it is the microphysical laws that determine which 
counterfactuals are true, that underwrite genuine explanations and that ground 
reasonable, well-informed, predictions. 

But such objections are misguided. The special scientific laws do not really 
compete in this way with the microphysical laws. Suppose, for example, that 
there is a fact of the matter about what the microphysics would have been had 
I crossed the pea plants and that the microphysical laws consequently generate 
a chance I for the cross resulting in a child homozygous for each trait. Then 
it is true both that 'if I had conducted a cross, then there would have been a 
chance I' and that 'if I had conducted a cross, then there would have been a 
chance 0.25' . Of course, these divergent chances cannot be the same chance. 
As I shall argue in Section 4, they are chances of different levels: the former is 
a micro-level chance, the latter a higher-level chance. 

In some cases only the latter counterfactual will be true. Suppose, for exam-
ple, that there is no fact of the matter about what the microphysics would have 
been had I crossed. Then it is nevertheless true in virtue of the Mendelian laws 
that 'ifI had conducted a cross, there would have been a chance 0.25'. 

Likewise, suppose that I am going to conduct a cross but it's still not set-
tled what the microphysical details of the cross will be, or I just don't know 
the microphysical details. Then I should have confidence of degree 0.25 in the 
prediction that a given resultant pea plant will be homozygous for each trait. 
Indeed, as I shall argue in Section 4.2 below, I should have this degree of cre-
dence even if the microphysical details are settled and I know all the admissible 
information about the case. 

As regards explanation, there may be more than one good explanation of 
an event. If a resulting pea plant turns out to be homozygous, then a good 
explanation would be that I conducted the cross and that the Mendelian laws 
indicate that there was a 0.25 chance that this outcome would result. This is 
true even though a still more satisfying explanation might be that I conducted 

9 Ifa I-Iumean analysis oflaws is correct, one might wonder whether laws are genuinely confirmable 
by their instances and, indeed, whether they can really support counterfactuals and underwrite 
explanations and predictions. But there is no special problem for special scientific laws here. The 
problem, if there is one, is with I-Iumeanism and not with special scientific laws. 



65 

the cross and the microphysical laws and circumstances entailed a chance I for 
this outcome. 

The special scientific laws play the law role, and ought therefore to be ac-
commodated as genuine laws by any adequate account of lawhood. And, in the 
previous subsection, I argued that there is good reason to think that they are 
accommodated by the Lewisian account. 

4 Deterministic Chance 

4.1 Chance and laws again 

In the previous section it was argued that there exist probabilistic special sci-
entific laws even in fundamentally deterministic worlds. When taken together 
with an initial history of the deterministic world in question, these laws entail 
non-trivial probabilities for the events that they concern. Since these laws are 
genuine objective laws, the probabilities that they project are genuine objec-
tive chances. Therefore, there exist fundamentally deterministic worlds with 
chance-projecting special scientific laws. 

In light of this, it is clear that Schaffer's assertion (noted in Section 2.3 
above) that the laws of deterministic worlds don 't project chances is just false. 
So Schaffer is wrong that non-trivial deterministic chances are incompatible 
with the connection from chance to lawhood, captured by his LMP: 

(LMP) If Chtw(p) = x, then the conjunction of Htw (a proposition giving 
the history of w up until time t) with the laws of w entails that Chtw(p) = x. 

The compatibility of non-trivial deterministic chances with the LMP can be 
seen by reconsidering the earlier example of the crossing of pea plants. 

Suppose that the world w at which the crossing occurs is a Newtonian 
one (the Mendelian laws are perfectly compatible with this). The history of 
w through I (the time of crossing)- which includes the fact that the crossed 
plants (Jim and Jill) are heterozygous for both shape and colour- together 
with the laws- which include the Mendelian laws-entail a chance 0.1875 for 
the proposition Prg, that Tom (the resulting pea plant) has round , green peas. 10 
Thus, we have I > Ch/w(Prg) = 0.1875 > O. 

But there is a difficulty here. Since w is a deterministic Newtonian world, 
the history of w through I- which includes the complete microphysical state 
of the world at I- together with the laws of w- which include the fundamen-
tal dynamical laws (alongside the Mendelian laws)-entails Ch/w(Prg) = I or 
Ch/w(Prg) = 0, depending on whether or not Prg is actually true at w . Thus, we 
have that I > Ch/ w(Pr g) > 0 and that Ch/w (Pr g) = I or O. Contradiction! 

10 In the case of pea crosses, genotypes a re reliably reflected in phenotypes. 



66 

Or so it seems: so long as it is maintained that chance is a function of just three 
arguments, there is a tension in assigning the same proposition two divergent 
chances at the same world and time. But one can escape the contradiction by 
allowing that chance isn't after all just a function of propositions, worlds, and 
times. 

Consider an analogy: for a given world and a given proposition, we may get 
more than one different chance. No one would regard this as a contradiction 
because chance is a function not just of worlds and propositions, but also 
of times. There is no tension between its being the case, for instance, that 
Ch,w(p) = x and Ch,.w(p) = y (where x =I y, t =I t'). Similarly, there is no 
tension between Ch,w(p) = x and Ch,w'(p) = y (where x =I y, w =I Wi). 

Likewise, if chance is a function of some fourth argument, then there would 
be no difficulty in admitting that there may be two or more divergent chances 
for the same proposition, in the same world even at the same time. 

What might this fourth argument be? The consideration that puts pressure 
on us to accept these divergent chances is the existence of laws of different 
levels that entail them. It seems, then, that there is an important level-relativity 
of chance, as well as its proposition-, world-, and time-relativity. 

One can capture this additional dimension of relativity by introducing an 
additional subscript to the chance function: 'Ch,wl(p), can be read 'the I-level 
chance of p at time t and world Wi. The I-level chance of p at t and w is just 
the chance entailed for p by the I-level laws of w when taken together with the 
history of w through t. 

In the above example, the Mendelian laws of w together with the history 
of w through t entailed that Ch,w(Prg) = 0. 1875 . The microphysical laws of 
w together with the history of w through t, on the other hand, entail that 
Ch,w(Prg) = I or O. But contrary to first appearances this is no contradiction, 
for the chances entailed by the Mendelian laws are not the same as those entailed 
by the microphysical laws: they are chances of different levels. Any appearance 
of contradiction evaporates when the appropriate indices are added to the 
chance functions. Let'" (h for higher) be the level with which the Mendelian laws 
are concerned, and let If (f forJundam entaf) be the microphysical level. Then, 
making the level-relativity of the chances explicit, we have Ch,wl" (Pr g) = 0.1875 
and Ch,wl/Prg) = I or O. No contradiction there! 

In general, it might be that Ch,wl,(p) = x, but that Ch,wl/p) = y (where 
x =I y, Ii =l l j ) . Divergent chances may exist for the same proposition at the 
same world and time because of the level-relativity of chance. Recognition of 
this fact is the key to the reconciliation of determinism with non-trivial chance. II 

II More generally, it is the key to the reconciliation of divergent chances proj ec ted by the laws 
of different leve ls. In th e special case of interest to us, where the world in qu esti o n is micro· 
deterministic, it allows a reconciliation of the trivial chances projected by the microphysical laws 
with non·trivi a l chances projected by high e r· level laws. 



67 

Whilst the LMP sanctions apparently inconsistent chances, relativizing 
chances to levels results in a statement of the chance-law connection that 
avoids this: 

(LMP') If Ch tw/(p) = x, then the conjunction of Htw (a proposition giving 
the history of w up until time t) with the I-level laws of w entails that 
Chtw/(p) = X . 

In a deterministic world that has probabilistic laws of non-fundamental levels, 
the original LMP sanctions non-trivial chances (as we saw earlier). The revised 
LMP* just makes it clear that these non-trivial chances are not the same as the 
trivial chances that are also sanctioned. Thus the appearance of inconsistency 
is avoided . Either way, given the existence of probabilistic special scientific laws 
in deterministic worlds, it is clear that non -trivial deterministic chances are 
compatible with the chance-law connection. 

Indeed, it seems that the conceptual connection between chances and laws 
might be tighter than that captured by either the LMP or the LMP·. These 
principles state roughly that all chances are lawfully entailed. But one might 
think, in addition, that all lawfully entailed probabilities for the occurrence of 
events are chances. 

If there is this closer connection between chances and laws, then taking only 
the values projected by Jundamentallaws as chances is incompatible with the 
chance-law connection. That connection now not only sanctions but also entails 
the existence of non-trivial chances in deterministic worlds with probabilistic 
special scientific laws and would be violated by a function that outputted only 
trivial chances in such a world. 

4.2 Chance and credence again 

Recall that Schaffer apparently showed that a non-trivial chance assignment 
to a proposition p in a deterministic world contradicts the deliverances of the 
reformulated Principal Principle: 

(RPP) 

The problem was that, if w is deterministic, then the laws and initial history 
will entail either p or ~p. Therefore, reasonable credence conditional upon the 
laws and initial history is equal to I or 0, from which it follows directly by the 
RPP that Chtw(p) is correspondingly equal to I or 0. 

Note that the reconciliation thus effected between chance and determinism is a re latively weak 
form of reconciliation: specifically, it is not claimed that determinism at a level I can be reconciled 
with non-trivial objective I-level chances. This point has as a corollary the observation (footnote 
7 above) that there may not exist non-trivial objective chances in thoroughly deterministic worlds: 
that is, worlds in which the laws of all levels are deterministic. 



68 

Note that it is the original PP rather than RPP that, according to Lewis, 
enjoys the 'direct intuitive support' ([ 1986b], p. 98). The P P was reformulated 
only because the reformulation 'provers] easier to use' (ibid.). Importantly, the 
demonstration of the incompatibility of non-trivial deterministic chances with 
the PP depends upon the validity of RPP as a reformulation of the PP. This is 
important because the reformulation is invalid. 

The original P P, it will be recalled, states that reasonable credence in p con-
ditional upon the proposition X that Ch/w(p) = x and any other proposition 
E compatible with X that is admissible at time t and world w, is equal to x: 

C(p I X.E) = x. (PP) 

The reformulated P P was arrived at by assuming that the big conjunction of 
the complete set of laws of w with the history of w through t is an admissi-
ble proposition at t and w. But this big conjunction is not, in general, fully 
admissible. 12 

Recall that Lewis characterizes admissible propositions as propositions 
whose impact on credence about outcomes comes entirely by way of credence 
about the chances of those outcomes. On the assumption (which Lewis makes) 
that there can only be one chance for a proposition in a world at a time, a 
chance that is entailed by the big conjunction of the initial history and the 
laws, it might seem that this big conjunction is indeed admissible. But this as-
sumption is incorrect: there exist laws of different levels which, taken together 
with initial history, entail divergent chances for the same propositions even in 
the same worlds at the same times. And, as will now be seen, the existence of 
these divergent chances makes it false that the big conjunction is in general 
admissible. 

Consider the high-level chance, entailed by the Mendelian laws, for the 
proposition Prg that Tom has round, green peas: Ch/wh(Prg) = 0.1875. Rel-
ative to this chance the conjunction, fitw.L(wh), of the history of w through t 
with the Mendelian and other high-level laws of w is admissible. It is admissible 
because its impact on reasonable credence in the proposition Prg comes entirely 
by way of its impact on credence about the value of Ch/ wl" (Prg). The proposi-
tion fitw.Lwl" entails Ch/w",(Prg) = 0.1875 and it contains no other information 
that is relevant to whether or not Prg is true. 

But it cannot be inferred from this that H,w.Lw, the proposition giving the 
history of w through t together with the complete set of laws ofw, is admissible. 
The problem is that this proposition may well carry information relevant to 

12 Lewis himself ([1994]) recogni zes that if a Humean view of laws is correct the laws are not fully 
admissible. He consequent ly acknow ledges that the RPP is not a strictly va lid reformulation of 
the P P. However, Lewis's reason for taking the laws to be strictly inadmissible centres around the 
problem of lIndel'l11iningjiillll'es and is somewhat different from that which is of concern here. It 
will be set aside in what follows. 



69 

whether Prg is true over and above that expressed by the proposition that 
Ch rw!" (p" g) = 0. 1875. In particular, Lw includes the laws of levels other than 
It, and these may project divergent chances for Prg' These chances constitute 
additional information relevant to whether Prg is true. 13 Indeed where w is 
fundamentally deterministic, Hrw.L w, which includes the fundamental laws, 
entails Prg or ~Prg and so clearly carries such additional information. 

Hence, relative to Chrw!,,(Prg), Hrw.Lw is simply inadmissible. And since the 
validity of the reformulation of the P P rested upon the assumption that Hrw . Lw 
is in general admissible, it can be concluded that the reformulation is invalid 
and that Schaffer's demonstration that non-trivial deterministic chances are 
incompatible with the PP, a demonstration that depended upon the validity of 
the RPP, is unsuccessful. 

This argument that RP P is invalid as a reformulation of P P and that therefore 
non-trivial deterministic chances can 't be shown to be inconsistent with the P P 
is one that has been made by Hoefer ([2007]). Regarding the alleged derivation 
of a contradiction from P P plus the supposition of a non-trivial deterministic 
chance for some proposition A, Hoefer says ([2007], p. 559): 

[T]his derivation is spurious; there is a violation of the correct understand -
ing of admissibility going on here. For if Hr w Lw entails A, then it has a big 
(maximal) amount of information pertinent as to whether A, and not by 
containing information about A's objective chance! So HrwLw, so under-
stood, must be held inadmissible, and the derivation of a contradiction fails . 

Whilst recognition of the level-relativity of chance makes it clear that the re-
formulation of the P P is invalid, it does not create any problem for the P P 
itself. The only necessary adjustment is that, once it is allowed that there may 
be more than one chance attaching to the same proposition at the same world 
and time, Lewis 's supposition of uniqueness must be dropped. Thus, X must 
not be read as the proposition that the chance at wand t of P is equal to x, but 
rather as the proposition that the I chance at wand t of p is equal to x, where 
I is a variable ranging over levels. What information is admissible will depend 
upon which level is in question. 

Lewis 's reformulation of the P P was intended to facilitate ease-of-use. 
Unfortunately, as has been seen, it involved the fallacious substitution of the 
not-generally-admissible proposition Hr w.Lw for the admissible proposition E. 
Can one give a user-friendly reformulation of the P P without committing such 
a fallacy? The key to doing so is (as with the LMP) to restrict the laws so as to 
be of the same level as the chances, as in (RPP*): 

(RPP*) 

13 This is a n a logo lls to th e manner in which the chances of P'g at times later than I cons titllte 
inadmiss ibl e additional information releva n t to whether P,g is true. 



70 

Intuitively: the initial history and I-level laws of the world entail the I-level 
chances and reasonable credence for someone whose evidence included all and 
only the initial history and I-level laws (and who learns by conditioning) is 
equal to the chance for p that they entail. 14 

Where I = II, the microphysical level, and w is a deterministic world, the 
chance entailed by Lwl. H,w will be trivial and reasonable credence conditional 
upon just Lwl. H,w will be correspondingly trivial. But where I is some non-
fundamental level, !", the chance entailed by Lwl . Htw need not be trivial and, 
if it is not, reasonable credence conditional upon just Lwl.Htw will be corre-
spondingly non-trivial. 

It would, of course, be interesting to know what reasonable credence would 
be for someone who knew the complete laws and initial history, Lw.H,w, and 
consequently the chances projected by the laws of each of the different levels. 
The RP p' , which is the correct reformulation of the P P, does not tell us- it 
does not indicate which of the divergent chances for p would guide rational 
credence in p . 

It does seem plausible that where w is a fundamentally deterministic world 
the answer is the trivial, fundam ental (level II ) chance of p , Ch twIJ(p).1 5 The 
reason for thinking this is that, in this case, Lw .H,w actually entails p or "'p. 
Still RP p', which is the correct reformulation of P P, does not yield this (or any 
other) answer. This isjust a limitation of RPP·. 

In any case, PP is not after all incompatible with non-trivial deterministic 
chances. It is only the incorrect reformulation RPP that is inconsistent with 
such chances. Once the reformulation is corrected, it becomes clear that the 
existence of non-trivial chances in deterministic worlds with probabilistic spe-
cial scientific laws is perfectly compatible with the connection from chance to 
rational credence. 

Indeed, it is possible to go on the offensive against the incompatibilist re-
garding chance and determinism. Once it is acknowledged that chances are 
level-rel ative, it can be seen that a chance function that outputted only trivial 
values in a deterministic world with probabilistic special scientific laws would 

14 Even this formulation is prob lematic if a Hum ea n view of laws is correct (and not just because 
of the problem of und ermining futures) . On a Hum ea n view the laws of all leve ls superve ne 
upon hi story. Co nseq uently, for some leve l f ' (01 f), the initial history I~w may constrain what 
the I' laws are. If so, so meon e who knew Htw would have information about what the /' laws and 
therefore the I ' chances are, and would thereby have information about wh eth er p is true that is 
not just information about the va lu e of CiItwt(p). The proposition I~w would therefore not ·be 
fully adm iss ibl e relative to Chtwt(p), and so eve n (RPP*) would not be a valid reformulati o n of 
(PP) . 

The so luti o n to thi s problem of inadmissibil ity mi ght be to restrict the hi sto rica l proposition to 
be a proposition (Ht wt) giving information only about th e hi story of w at level I. Such propositions 
are discussed further in the Appendix. 

15 Ho efer ([2007] , pp. 592- 3) a rgues that in fundam entally indeterministic world s reasonable cre-
dence needn't track th e fundamental level chances in cases ofconftict. 



71 

not be one that outputted values that play the role of chance in guiding rational 
credence captured by the PP (and its valid reformulation, RPP*). 

For example, where h is the level of concern to the geneticist and w is 
a deterministic world with Mendelian genetic laws, a chance function that 
assigned a level h chance I or 0 at the time at which Jim and Jill are crossed 
to the proposition that Tom will have round , green peas would be one that 
assigned values that fail to guide rational credence given all of the relevant 
admissible information, information that includes just the initial history plus 
the I" laws (among which are the Mendelian laws but not the microphysical 
laws). Reasonable credence given just this information is 0.1875. 

Consequently, not only is it not the case that non-trivial deterministic chance 
functions are incompatible with the chance~redence connection, it is in fact the 
case that only such a function can underwrite this connection in deterministic 
worlds with probabilistic special scientific laws. In the previous section, we saw 
that the same was true with respect to the chance- law connection . 

4.3 Chance and possibility again 

It will be recalled that Schaffer argues that non-trivial deterministic chance 
functions are incompatible with the connection from chance to possibility 
captured by the RP: 

(RP) If Chtw(p) > 0, then there exists a world w' such that (i) w' matches 
w in laws; (ii) w' matches w in occurrent history up until time t; (iii) P is 
true at w'. 

As was seen in Section 2.2 above, if w is a deterministic world and Pe is the 
proposition that some event e occurs, then the initial history and laws of w will 
entail either Pe or ~ Pe (depending on whether e occurs in w). Accordingly, a 
non-trivial chance assignment to Pe is incompatible with the RP. 

Note that the formulation of the RP presupposes that the chance func-
tion yields a unique chance assignment for a given P, t , and w. But I have been 
arguing that this supposition is erroneous: chances are level-relative and, conse-
quently, there may be two or more divergent chances attaching to a proposition 
at a world and a time (these chances being chances of different levels). And 
note that the RP will be compatible with deterministic chance provided that 
the laws in question are restricted so as to be of the same level as the chance. 
Thus consider RP*: 

(RP*) If Cht w/(p) > 0, the n there exists a world w' such that (i) w' matches 
w in I-level laws; (ii) w' matches w in occurrent history up until time t; 
(iii) p is true at w' . 



72 

Where If is the microphysical level, and w is a deterministic world in which 
event e fails to occur, then Lwl/. Htw will entail ~ Pe. There is therefore no world 
Wi at which LWlr Htw .Pe is true. So, by RP' , Ch/wl/Pe) = O. If, on the other 
hand, e does occur at w then analogous considerations show that RP' implies 

Ch/wl/ (Pe) = I. 
But, where It, is some non-fundamental level, Lwlt, .Htw needn't entail ei-

ther Pe or ~ Pe, but may instead entail some non-trivial chance for Pe. If so, 
there are some worlds at which Lwlt,. H/ w. Pe is true and some worlds at which 
Lwlt, . Htw. ~ Pe is true. So it is consistent with RP' that I > Ch/wlt,(Pe) > O. 
Nor is there a contradiction between its being the case that I > Ch/wl,,(Pe) > 0 
and its being the case that Ch/wl/Pe) is equal either to I or O. This is because 
Ch/wlt, (Pe) and Ch/wl/ (Pe) are not the same chance, but are chances of different 
levels. 

One might insist that high-level deterministic 'chances' aren't really chances 
since they merely fulfil RP' and not RP, which (it might be insisted) is the correct 
chance- possibility principle. In order to evaluate this objection, it is necessary 
to consider the nature of the platitude concerning chance and possibility, of 
which RP and RP* are both candidate precisifications. 

Let us suppose that Schaffer (p. 124) is correct when he says that the platitude 
is roughly that 'if there is a nonzero chance of p, this should entail that P is 
possible, and indeed that P is compossible with the circumstances'. 

Note, first of all, that non-trivial deterministic chances aren 't supposed to 
be chances for events that are impossible in the sense that they are ruled out by 
the laws alone (as would be the case if, in our world, a non-trivial chance were 
assigned to the transmission of a super-luminal signal) . But it might be said 
that they are chances for events that, given the laws, are not compossible with 
the circumstances, including the microphysical initial history of the world . 

But, we might ask: which laws? In a fundamentally deterministic world , 
the events in question may be ruled out by the fundamental laws, given the 
circumstances. But, as has been seen, they need not be ruled out by the non-
fundamental laws. RP' allows t hat the I-level chance of an event e may be 
positive provided that e is not, in the circumstances, ruled out by the laws of 
that level. 

The objector might still insist that the chance- possibility connection is cap-
tured by RP, which rules out deterministic chances because the events in ques-
tion are not compossible with the circumstances given all the laws, rather than 
RP', which allows them because the events they concern are compossible with 
the circumstances plus the higher-level laws. 

The dispute is then one over the precise content of the chance- possibility 
platitude. And one might wonder whether this content is rich enough to allow us 
to adjudicate between RP and RP* as candidates for the correct precisification 
of the chance- possibility connection. 



73 

Indeed, these aren't the only two contenders. Schaffer ([2007], p. 124) takes 
the RP to be a relatively uncontroversial strengthening of Bigelow, Collins, and 
Pargetter's Basic Chance Principle ([1993] , p. 459): 

(BCP) If Chtw(p) > 0, then there exists a world Wi such that (i) Chtw'(p) = 
Chtw(p); (ii) Wi matches w in occurrent history up until time I; (iii) p is true 
at Wi . 

But there is evidently no inconsistency between BCP and non -trivial determin-
istic chances. Suppose that w is microphysically deterministic, and that Wi is 
not (the BCP doesn't require wand Wi to agree in laws). And suppose that both 
worlds are Mendelian, that Jim and Jill are crossed at time t in both worlds, 
and that Prg is the proposition that Tom (a resulting pea plant) has round, 
green peas. Then Chtwlt,.(Prg) = Chtwt,.(Prg) = 0. 1875. Since W is microphysi-
cally deterministic, it might be that Tom is determined to have wrinkled, yellow 
peas in w. It is compatible with this that Tom turns out to have round, green 
peas in Wi (since Wi is not microphysically deterministic). Therefore, the BCP is 
compatible with the existence of a positive chance for Prg in w, even though Prg 
is incompatible with the microphysical laws plus microphysical history of w . 

Nevertheless, suppose BCP to be somehow ruled out as a candidate pre-
cisification of the chance- possibility connection, and take just RP and RP* 
as the relevant alternatives. There are the following grounds to favour RP* 
(which is compatible with non-trivial deterministic chances) over RP (which is 
not). 

If, as it seems one must when discussing the content of platitudes about 
chance, one takes seriously the layperson's talk about chances for ordinary 
macroscopic events- such as the 50% chance of the coin landing heads- or 
the scientist's talk about chances for special scientific events- such as the 35% 
chance of the hurricane making landfall in southern Florida- then there is 
pressure to conclude that chances are not the sorts of things that must (as the RP 
requires) indicate compossibility with the circumstances given thejimdamen tal 
laws. The layperson and the meteorologist may not know, or have any opinion 
about, whether these events are compossible with the circumstances given the 
Jundamentallaws. Yet any doubt about this does not lead to doubt about the 
correctness of these chance assignments. 

What the layperson does know is that (as required by the RP*) heads is not 
ruled out by the macro-level law ascribing a 0.5 chance to coin flips . What 
the meteorologist knows is that (again, as required by the RP*) the south 
Florida landfall is not ruled out by her model, a model that encodes (at least 
approximations of) the probabilistic functional laws of her discipline. 

There is another reason to favour the RP* over the RP. Change the ear-
lier example so that the crossing of the pea plants now occurs in a micro-
indeterministic world, Wi. And suppose that the Mendelian laws still hold at 



74 

Wi, SO that Ch/w;l" (~Prg) = 0.8125. Suppose, moreover, that both Prg and ~ Prg 
are compossible with the circumstances plus the fundamental laws of Wi, so 
that this chance is consistent with the original RP. 

But suppose that ~ Prg is only compossible with the fundamental laws in 

virtue of the possibility of an extremely unlikely quantum event,16 so that the 

chance assigned by the fundamental laws to ~ Prg is exceedingly small, e.g., 

Ch/w;l/~ Prg) = 10- 1.000 ,0°°. Surely it is bizarre in the extreme to maintain that 
it is this immensely unlikely and seemingly irrelevant possibility that grounds 

the positive value of Ch/w;it. (~Prg) . Remove the astronomically small quantum 

uncertainty, and suddenly the value of Ch/w;l" (~Prg) switches from 0.8125 to 

o (despite the continued existence of the 0.8125 probability projected by the 
Mendelian laws)! This is hardly plausible. 

The'" chance of 0.8125 sanctioned by the RP* clearly remains even if the 
fundamental dynamics of the world turns out to be deterministic. The elimina-

tion of the very slight quantum uncertainty has only the effect of changing the 
value of the If chance from 10- 1,000,000 to 0, again in accordance with the RP*. 

So the RP* looks like a better chance- possibility principle than RP, and since 
the former sanctions non-trivial chances in deterministic worlds with proba-

bilistic special scientific laws, it seems that non-trivial deterministic chances are 

compatible with the chance- possibility connection. 

5 Chance and Causation 

Schaffer claims that non-trivial deterministic chances would violate the connec-

tions from chance to rational credence, possibility and lawhood. In the previous 
section it was shown that recognition of the level-relativity of chances makes it 

clear that this is not so. Indeed it was argued that in deterministic worlds with 

probabilistic special scientific laws only a non-trivial function can underwrite 

the chance- credence and chance- law connections. 

The aim of this section is to reinforce the conclusion that there are de-
terministic chances by considering one of the three remaining platitudes that 

Schaffer identifies: namely, that connecting chance to causation. Schaffer says 

that he sees no incompatibility between the chance-causation platitude and a 

non-trivial deterministic chance function (he says the same about the final two 

platitudes, chance- futurity and chance- intrinsicness). But I shall argue that 

there is an incompatibility with trivial deterministic chance functions. 

The argument is independent of considerations concerning the level-relativity 

of chances. Consequently, in what follows, the level index to the chance function 

will be suppressed for simplicity. The argument is also ancillary to the main case 

16 This might be Railton 's ([19781, p. 224) fanta stic eve nt in which 'a ll the naturally unst able nuclid es 
on earth [ .. . 1 commenced spo ntaneous nuclear fission in rapid succession'. 



75 

for deterministic chance made in Sections 3-4 above, which goes through even 
if one rejects the claim that there is a platitudinous connection from chance to 
causation (or if one rejects the admitted ly contentious precisification of that 
platitude that I suggest below). 

Schaffer ([2007], p. 126) claims that the platitude about the connection from 
chance to causation is as follows: 

[Clhances shou ld live within the causal transitions they impact. That is, 
if a given chance is to explain the transition from cause to effect, that 
chance must concern some event targeted within the time interval from 
when the cause occurs, to when the effect occurs. Otherwise that chance 
cannot impact the transition from cause into effect- it would be left outside 
of the action. 

He states (ibid.) that this platitude may be codified as the causal transition 
constraint (CTC): 

(eTC): If Ch tw(P.) plays a role in the causal relation between c and d, then 
teE[te, t,il.J7 

Schaffer (p. 132) says that he sees 'no problem' about non-trivial deterministic 
chances fitting the CTc. 18 

I am sceptical about the status of this 'platitude'. The phrase 'chances should 
live within the causal transitions they impact' seems very obscure, as does the 
notion of a chance's being able to 'explain the transition from cause to effect' . 
Even the notion of a chance's 'playing a role in a causal relation', which appears 
in the alleged precisification of the chance-causation platitude, CTC, is very 
opaque. 

Insofar as I can make sense of these statements, they just seemfalse (and can 
therefore hardly be platitudinous). Consider the following example. 19 Suppose 
that Napoleon is on his way to Waterloo. I am a general in the Prussian army and 
set an ambush . My men attack Napoleon's troops, inflicting heavy losses (e). 
Napoleon reaches Waterloo with his army severely depleted and unable to 
withstand Wellington 's charge (c). Wellington is victorious (d). 

Let Ch,,_,:I,,@(Pe) be the chance, just before I attacked, of my inflicting heavy 
losses on Napoleon. It seems that this chance helps explain why I did in 
fact inflict heavy losses upon Napoleon (e) . This, in turn, helps explain why 
Wellington's charge (c) resulted in his victory (d). There does not seem to 
be any failure of transitivity here, and so (insofar as I can make sense of these 

17 ' e , /d, and /e here denote the times at which the subscripted events occur. 
18 At least non-trivial deterministic chances of the posterior sort with which this paper is concerned 

(see footnote 2). 
19 The example is due to John Hawthorne, who gave it at a seminar in Oxford on Schaffer's paper. 

Y have embe lli shed it somewhat. 



76 

notions) it seems that the chance of e helps explain (and 'impacts' upon) the 
causal transition from c to d. Nevertheless, te rt [te, tel] . Thus, insofar as the 
CTC is meaningful, it seems to be false. 

In any case, it strikes me as implausible to hold that the claim 'chances [ ... ] 
live within the causal transitions they impact' is the obvious candidate for the 
platitude connecting causation and chance. I think many more people have 
thought the following to be a platitude: 'causes (tend to) raise the chance of 
their effects'. 20 

A natural way to be precise about the relevant chance-raising relation is to 
cash it out in terms of an inequality between two conditional chances: 

(I) 

This inequality says that, just before c occurred, the chance of e conditional 
upon the occurrence of c was greater than the chance of e conditional upon the 
non-occurrence of c. 

The result is the following chance-causation constraint (CCC): 

(ccq If c is a cause of e then (ceteris paribus) c raises the chance of e in 
the sense that (1) obtains. 21 

Since chances obey the probability calculus (see Lewis [1986b], esp. p. 98) we 
have, by the axiom of conditional probability: 

Ch ( ) _ Chtw(p.q) 
lw P ! q - Chtw(q) , Chtw(q) > 0, (2) 

Where Chtw(q) = 0, the axiom of conditional probability leaves Chtw(p!q) 
undefined . 

Suppose that w is deterministic and c is a cause of e. Then, by CCC, it is the 
case (ceteris paribus) that c raises the chance of e in the sense that (I) holds. But 
(I) will hold only if both terms are well defined. By (2) (plus the Complementa-
tion Theorem) both terms will be well defined only if I > Cht, - ew(Pe) > 0. But, 
by the assumption that there are only trivial chances in deterministic worlds, 

20 This has been a principal motivation for those (such as Good [196Ia] , [196Ib] , [1962]; 
Reichenbach [1971]; Suppes [1970] ; Lewis [1986d] ; Menzies [1989]; Eells [1991]; and Kvart [2004]) 
who have attempted to develop probabilistic analyses of causation. Even those who don't seek a 
probabilistic analysis tend to agree. For instance, Mellor ([1995], esp. p. 67) takes th e tendency 
for causes to raise the chance of th eir effects to follow from the ' connotations of causation'. 

21 The level·relativity of chance makes for further subtleties in the precisification of the chance-
causation connection . In particular there is the question of which le"el~' chance of e must (ceteris 
paribus) be raised by c if c is to be a cause of e. Such subtleties are rather similar to those arising 
from the time-relativity of causation (inequality (I) incorporates the implicit assumption of one 
possible- and somewhat vague- answer to the question of which time's chances are relevant to 
whether c is a cause of e) . 



77 

it follows that Ch,c-'w(Pc) is equal to I or O. Contradiction! Conclusion: a 
chance function that outputs only trivial chances in deterministic worlds is 
incompatible with the connection between chance and causation captured by 
the CCc. 22 

Note that, although CCC says only that causes raise the chance of their 
effects ceteris paribus, a chance function that outputted only trivial values in 
deterministic worlds would mean that causes never raise the chance of their 
effects in those worlds. By CCC, this is unacceptable. 

6 Conclusion 

The probabilistic special scientific laws of deterministic worlds, when taken 
together with the initial histories of those worlds, entail non-trivial probabilities 
for the events that they concern. In addition to being lawfully projected, these 
non-trivial probabilities guide rational credence and indicate compossibility with 
the special scientific laws plus circumstances. They therefore play the chance 
role and should be considered chances. 

Indeed, not only are there non-trivial deterministic probability functions (de-
terministic probability functions that output at least some non-trivial values) 
that play the chance role, there are such functions that play it better than any 
trivial deterministic probability function. In deterministic worlds with proba-
bilistic special scientific laws, only a non-trivial probability function can under-
write the connections from chance to lawhood and rational credence (as was 
argued in Sections 4.1 and 4.2). And if the argument of Section 5 is found com-
pelling there is reason to think that only a non-trivial function can underwrite 
the connection from chance to causation. 

The conclusion that there exist non-trivial deterministic chances is not in 
tension with the claim that there also exist trivial chances (trivial probabilities 
that also play the chance role) for the very same propositions at the very 
same times and worlds (outputted by the very same chance function) . Any 
appearance of tension dissolves once it is recognized that chance is a function 
offour arguments: a proposition, a time, a world, and a level. 

The illusion of tension between trivial and non-trivial deterministic chances 
has been the reason for opposition to the latter. This is in evidence in the 
passage quoted from Lewis in Section I. It is also in evidence in Schaffer's 
arguments (reviewed in Section 2) that sought to derive a contradiction from 
the supposition of non-trivial deterministic chances. 

22 Caveat: Hiljek ([2003a], [2003b], [2007]) has- in my view rather convincingly- challenged the 
validity of the axiom of conditional probability. Since the above demonstration ofincompatibility 
between the CCC and a trivial deterministic chance function depends upon that axiom , it will 
(at least in its present form) not be found entirely convincing by those who endorse Hajek 's 
arguments. 



78 

Chance can be reconciled with determinism, and disparate chances can be 
reconciled with one another. The reconciliation is effected by the level-relativity 
of chance. Lewis is correct in 0 bserving (in the passage quoted in Section I) that 
it is false that the chance of a fair coin landing heads can be both zero or one and 
also 50%. But it is only false because the supposition of uniqueness is false. It 
can be the case both that a coin has a chance zero or one and a chance 50%: the 
chances being chances of different levels. Once the illusion of tension between 
disparate chances is dispelled through recognition of chance's level-relativity, 
there remains no obstacle to accepting that the non-trivial, chance-role-playing 
probabilities projected by the special scientific laws of deterministic worlds are 
indeed deterministic chances. 

Appendix: Times, Levels, and Chance Setups 

Hajek ([2003a], [2003b], [2007]) has convincingly argued that chances must be 
relativized to chance setups. This seems to be implicitly accepted by both Lewis 
and Schaffer. Both relativize chances to times , but as Hoefer observes ([2007], 
pp. 564-5):23 

For Lewis, a non-trivial time-indexed objective probability Chtw(P) is, in 
effect, the chance of p occurring given the instantiation of a big setup: the 
entire history of the world up to time 1.24 

This suggests that the following analysis might be adopted: 25 

Chtw(p) =def Chw(p I Htw). 

Chances, on this analysis, are not fundamentally time-relative. Rather, the 
chance of p at t and w is just the chance for p that results from conditioning 
upon the history of w through t. This analysis has its merits: if p concerns what is 
past by t then Htw entails p and Chw(pi Rtw) = I; moreover, since the big chance 
setup Rtw changes as the value of t changes, so will the chance distribution that 
results from conditioning upon it. In short, conditioning the chance function 
on history through time t reproduces the properties of a time-indexed chance 
function. 

I argue in the paper that chances are level-relative. In Section 3.2, it was 
suggested that the relationship of realization grounds the distinction between 
higher and lower levels: two sciences can be regarded as (at least partially) 
characterizing distinct levels when various distributions of the properties of 
concern to one realize those of concern to the other. 

23 I have modifi ed Hoefer's notation to render it consis tent with my own. 
24 This observation applies just as well to SchatTer as it does to Lew is. 
25 Eagle ([unpublished]) gives such a n analysis. 



79 

With this in mind, define the history of world w at level I up until time t as 
the distribution of I-level properties throughout w up until t. The proposition 
giving this distribution can be written' lftwi'. I think that the following analysis 
might then be adopted: 

Ch/w/(p) =def Chw(p I lftw/)' 

Chances, on this analysis, are not fundamentally level-relative. The I-level 
chance of p at t and w is just the chance for p that results by conditioning 
upon the I-level history of w through 1. 26 This analysis has its merits, foremost 
among which is that if the I-level history together with the laws of w don't 
entail p, then conditioning upon lftw/ won't trivialize the chance of p. Condi-
tioning the chance function on I-level initial history therefore reproduces this 
important property of a level-indexed chance function. 

Acknowledgements 

For helpful discussion and comments on earlier versions of this paper, I'd 
like to thank Frank Arntzenius, Roman Frigg, Marion Ledwig, Miklos Redei, 
Jonathan Schaffer, Mauricio Suarez, Alistair Wilson, and two anonymous ref-
erees of this journal. I would also like to thank audiences at several presentations 
of this paper in 2008, including at the British Society for the Philosophy of Sci-
ence Annual Conference, the Joint Session of the Aristotelian Society and the 
Mind Association, the LSE Philosophy of Probability Graduate Conference, 
the Ockham Society at Oxford, and the Philosophy of Probability Research 
Seminar at King's College, London. I'd like especially to thank Antony Eagle 
for detailed comments on several earlier drafts of this paper. Funding for this 
research was provided by an Arts and Humanities Research Council Doctoral 
Award. 

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