The Correlation Argument for Reductionism

Christopher Clarke∗

Forthcoming in Philosophy of Science

Abstract

Reductionists say things like: all mental properties are physical properties; all
normative properties are natural properties. I argue that the only way to re-
sist reductionism is to deny that causation is difference making (thus making
the epistemology of causation a mystery) or to deny that properties are indi-
viduated by their causal powers (thus making properties a mystery). That is
to say, unless one is happy to deny supervenience, or to trivialize the debate
over reductionism. To show this, I argue that if properties are individuated
by their causal powers then, surprisingly, properties are individuated by nec-
essary co-exemplification.

1 Property Monism versus Property Dualism

Some properties are social properties such as is the president of Columbia; some prop-
erties are mental properties such as likes broccoli; some properties are biological
properties such as has a heart or is a mammal; some properties are physical prop-
erties such as has high temperature; and indeed some properties are microphysical
properties such as contains ammonium ions. Or so it is often said. But what is the
relationship between these various domains of properties? Are all social prop-
erties in some sense mental or biological properties (Mellor 1982; Pettit 1993;
Ruben 1984; Sawyer 2002)? Are all mental properties in some sense biological or
physical properties (Smart 1959; Lewis 1966; Chalmers 1996)? Are all biological

∗Thank you Adrian Boutel, Christopher Cowie, Hugh Mellor and Mark Pindar for your com-
ments on an ancestor of this manuscript; and also to several anonymous referees for your generous
and constructive suggestions. This work has received funding from the European Research Coun-
cil (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant
agreement 715530).

1



properties in some sense physical properties (Rosenberg 2006; MacDonald 1992)?
Are all physical properties microphysical properties (Pettit 1993; Papineau 2001;
Hüttemann 2004; Mellor 2008)? For that matter, are all normative properties in
some sense natural properties (Streumer 2008)?

Thus for any two domains of properties one can ask: is one domain a subdo-
main of the other? Property monists say yes, and property dualists say no. Of
course, these are relative designations: one might be a monist about social prop-
erties in relation to mental properties, for example, but a dualist about biological
properties in relation to physical properties. (Another complication: so-called
non-reductive physicalists are monists about all properties in relation to ‘broadly
physical’ properties, but they are dualists about all properties in relation to ‘nar-
rowly physical’ properties.1 And so, in examining an argument against all vari-
eties of dualism, this paper will be examining an argument against non-reductive
physicalism.)

Claims of monism and dualism often figure as key premises or considerations
in debates about laws and explanations in biology, psychology, social science.
What is the nature of biological, mental and social scientific laws? Can the laws
and facts of one domain be fully explained by invoking the laws and facts of an-
other? Can explanations that invoke concepts from one domain nevertheless be
reformulated solely in terms of the concepts of another, and without loss of under-
standing (Clarke 2016; Clarke 2017)? Although this paper will not itself explore
these questions about laws and explanations, the importance of these questions is
part of what makes the issue of monism versus dualism so important itself. (Never-
theless, one ought to keep all these questions separate. To do so, I will talk about
property monism versus property dualism, rather than about reductionism versus
anti-reductionism. This is because talk of reductionism versus anti-reductionism
blurs together the above questions about laws and explanations with the present
question about properties.)

Within the philosophical literature, one finds three arguments that aim to es-
tablish monism over dualism. Firstly there are those arguments that appeal to
Kim’s infamous exclusion principle, or something similar such as filler function-
alism.2 This paper will not address such arguments. Secondly there are the argu-
ments that appeal to Kim’s inheritance principle.3 This paper will also not address

1A property is broadly physical if the property is ‘determined’ or ‘realized’ by the narrowly
physical properties (Pereboom and Kornblith 1991; Shoemaker 2001). What counts as a narrowly
physical property, is however, notoriously difficult to define (Ney 2008).

2See Kim (1998, §2) for explicit exposition. See Lewis (1966, 1994) and Jackson (2002) for a
variant that appeals also to filler functionalism, and which implicitly invokes the exclusion princi-
ple. See Bennett (2003) and Macdonald and Macdonald (2006) for critical responses.

3See Kim (1998, 107–11) for an exposition and defence of the argument itself. Refer also to
Kim (1992), Lewis (1994), Shapiro (2000), Heil (2003), and Heil and Robb (2003). See Pereboom

2



such arguments. Thirdly there are various arguments that I will call correlation
arguments.4 And one of the most interesting versions of the correlation argument
is that given by Antony (2008).

The most prominent premises of Antony’s version of the correlation argument
are: () the properties of the first domain in question ‘supervene’ on the prop-
erties of the other domain in question, and (  ) properties are indi-
viduated by ‘necessary coexempfication’ (section 2). Many philosophers warmly
embrace , many dualists included. But the status of    is less
clear (section 3). So the correlation argument, as it stands, is not as forceful as it
might be.

This paper aims to make the correlation argument more incisive and powerful,
and thereby to clarify the fundamental points that are at issue between monists
and dualists. Section 2 gives my preferred version of the correlation argument,
a version that comprehensively lays bare all the argument’s commitments. In
particular, I show how the argument can remain neutral on the existence and ob-
jectivity of multiply realizable properties and of disjunctive properties. Sections 3
and 4 will then show that    follows from some broadly appealing
theses about the metaphysics of causation and of properties. The most prominent
of these theses are: (  ) properties are individuated by their causal
powers, and ( ) singular causation is difference making. Sec-
tion 5 shows that another assumption of the correlation argument follows from a
plausible thesis about the semantics of ‘the property of ’ locutions. Section 6 sum-
marizes the achievement: there are four main ways in which a dualist can resist
the correlation argument. Dualists can give up at least one of three metaphysi-
cal theses—  , or  , or —each of which
currently enjoy widespread acceptance amongst dualists. Or dualists can make
their disagreement with monists a superficial or semantic disagreement: over what
properties to label as physical, for example. Section 7 reviews the costs of each of
these options.

Thus I hope to deepen our understanding of dualism’s prospects, and to push
the debate forward by putting pressure on each dualist to be explicit about which
one of these four ways of resisting the correlation argument she will opt for.

and Kornblith (1991), Shoemaker (2001) and Baker (2007, 115–7) for critical responses.
4See Smart (1959) and McLaughlin (2007). Bacon (1986), Kim (1992), Clapp (2001) and

perhaps Jaworski (2002) may also be interpreted as offering a variant on this argument. Some
correlation arguments appeal to considerations of simplicity (Smart 1959) or inference to the best
explanation (McLaughlin 2007) to argue that it’s rare for two distinct properties to be necessarily
co-exemplified.

3



2 The Correlation Argument

What’s the most compelling version of the correlation argument? Let’s begin with
a very sketchy example of a correlation argument in action.

The following holds necessarily: for any given object, that object is alkaline if
and only if it contains hydroxide ions or it contains ammonium ions. Or so I will
pretend for ease of illustration. But, just for the moment, let’s assume that the con-
cept is alkaline succeeds in denoting a property. From this one infers that there also
exists a property contains hydroxide or ammonium ions. But, one contends, this property
counts as a microphysical property. So the property is alkaline is necessarily co-
exemplified with the microphysical property contains hydroxide or ammonium ions. (By
saying that a property A and a property B are necessarily coexemplified I mean: it
is absolutely impossible for any object to exemplify property A without that object
also simultaneously exemplifying property B; and it is absolutely impossible for
any object to exemplify property B without that object also simultaneously exem-
plifying property A.) From this one infers that the property is alkaline is the very
same thing as the microphysical property contains hydroxide or ammonium ions. The
property is alkaline therefore counts as microphysical. One can now discharge the
temporary assumption that the concept is alkaline denotes a property. This yields:
if the concept is alkaline denotes any property at all, then the property it denotes
counts as microphysical.

Although the above argument is extremely sketchy, it does give a rough il-
lustration of my preferred version of the correlation argument, as applied to the
concept is alkaline, and in relation to the domain of microphysical properties. I will
now make the argument’s assumptions fully explicit, and I will restate the argu-
ment in its general form, for arbitrary concept a, and in relation to an arbitrary
domain of properties upon which concept a is assumed to supervene.

Consider those properties S1, S2, S3 . . . that belong to the arbitrary domain
of properties S in which we are taking an interest. This might be the domain of
microphysical properties, as it was in the example above, or it might be the do-
main of narrowly physical properties that I mentioned in section 1, or it might be
the domain of natural properties—whatever you like. Now, one can ask of any
object: what ‘total conjunction’ of these S-properties does that object exemplify?
For example, in the simple case in which S contains only three properties, eight to-
tal conjunctions of S-properties are possible—one example conjunction being the
conjunction in which S1 is exemplified but S2 and S3 are not exemplified. We can
also ask for any total conjunction of S-properties whether this conjunction is com-
patible with concept a. For example, since it is possible for there to be an object
that is alkaline, and which at the same time exemplifies precisely those properties
given by the the conjunction contains hydroxide ions and does not contain ammonium ions,

4



the latter conjunction is alkaline-compatible. More generally, to say that a con-
junction is a-compatible is to say that it is possible for there to be an object that
is [insert concept a here], and which at the same time exemplifies precisely those
S-properties given by this conjunction.

With these definitions in hand, let’s assume

() The following is absolutely necessary: whenever any object exem-
plifies precisely those properties given by any a-compatible total conjunction
of S-properties, then that object is [insert concept a here].

For example, it is absolutely necessary that, whenever any object exemplifies those
properties given by the conjunction contains hydroxide ions and does not contain ammonium
ions, then that object is alkaline. In this respect,  says that this domain of
S-properties determines whether or not concept a applies to any given object;
the latter concept supervenes on the former properties. (To be clear: to say that
something is absolutely necessary is to say that it would be absurd if this were
not the case. In other words, ‘the [absolutely] necessary is that whose negation
counterfactually implies a contradiction’ (Williamson 2007, 157). See section 6 for
discussion of some understandings of supervenience that do not appeal to absolute
necessity.)

But let’s also assume

( ) For any two concepts x and y, there is a concept x and y
that is their conjunction, and a concept x or y that is their disjunction, and
there is a concept that is their negation, not x for example.

There is a concept, for example, that is the disjunction of the concept contains
hydroxide ions and of the concept contains ammonium ions. One might call this concept
the concept contains hydroxide or ammonium ions. To be more precise, it’s a concept for
which the following is knowable a priori: an object is [insert this concept] if and
only if this object contains hydroxide ions or this object contains ammonium ions.

It follows from   that there is a concept that is the disjunction
of all the a-compatible total conjunctions of S-properties; call this concept b. (For
example, at the beginning of this section, I took the concept contains hydroxide or
ammonium ions to be the disjunction of all alkaline-compatible total conjunctions of
microphysical properties.) But it follows from  that concept b is necessarily
co-applicable with concept a.5 By necessarily co-applicable I mean: it is absolutely
necessary that any given object is [insert concept a here] if and only if this object
is [insert concept b here].

But let’s assume—temporarily for conditional proof—that concept a denotes
a property. Call this property A. And let’s assume

5See Kim (1984, 169–70), Bacon (1986) and Kim (1991, 151) for proof.

5



() If a concept denotes a property, then any necessarily co-applicable
concept itself denotes some property.

So concept b denotes some property too. Call this property B. So B is necessarily
coexemplified with A. But let’s also assume

( ) Take any class of concepts, all of which denote proper-
ties of type S. And take any concept that is formed purely out of taking con-
junctions and disjunctions and negations of the concepts in this class. Then
if this latter concept denotes a property, this property is itself of type S.

For example, if a property is denoted by the concept contains hydroxide or ammonium
ions, then this property is a microphysical property, because the concept contains
hydroxide ions denotes a microphysical property and the concept contains ammonium
ions denotes a microphysical property.

It follows that property B is an S-property. But let’s assume that properties are
individuated by necessary co-exemplification:

(  ) If any property P and any property Q are necessarily coex-
emplified, then P is the very same property as Q.

So property A is the very same thing as S-property B. So property A is an S-
property. We can now discharge our temporary assumption that concept a de-
notes a property. This yields: if our given concept a denotes a property, then this
property is an S-property.

I do not suggest that the above argument should compel everyone to accept
monism. Most of the assumptions in the above argument can be reasonably dis-
puted. That said, section 5 will lend additional and independent support to -
, and sections 3 and 4 to   . For the moment, however, I want to
pre-empt some objections by clarifying three things.

Firstly,   is very liberal about the existence of concepts. It is-
sues in a concept is either negatively charged or made of iron, for example. But 
 does not say that all concepts enjoy the same metaphysical status, as it
were. Perhaps the concept is either negatively charged or made of iron does not denote
a property, whereas the concept is made of iron does (Heil 2003). Or perhaps the
concept is either negatively charged or made of iron does denote a property, but the prop-
erty it denotes is not an objectively important property: it is a ‘gerrymandered’ or
‘disunified’ property, whereas is made of iron denotes a ‘natural’ or ‘unified’ prop-
erty (Fodor 1974; Lewis 1983). Thus   is compatible with the
idea that some concepts (is either negatively charged or made of iron for example) have an
objectively inferior status to others (is made of iron for example).

Secondly, my way of formulating the correlation argument does not specify
the status of concept a (the concept is alkaline for example). When my argument

6



assumes that is alkaline denotes a property, it does so only temporarily for condi-
tional proof. Thus my argument remains neutral about the existence or objective
importance of ‘multiply realizable’ properties. Similarly, my argument does not
specify the status of concept b (the concept contains hydroxide or ammonium ions for
example). Thus my argument remains neutral about the existence or objective
importance of ‘disjunctive properties’. Instead, what my argument commits to is
, which is very roughly speaking the claim that the concept is alkaline de-
notes a property if and only if the concept contains hydroxide or ammonium ions denotes
a property. It is of the utmost importance to note, however, that  on its own
does not entail that the concept is alkaline denotes the same property as the concept
contains hydroxide or ammonium ions, for example. Otherwise assuming  would
trivialize the correlation argument by making the other premises in the argument
redundant. Instead,  merely entails that if the former concept denotes a
property, then this latter concept denotes some property or other as well; see sec-
tion 5 for discussion of this point. At any rate,  is a controversial thesis,
I anticipate, and for this reason section 5 will lend additional and independent
support to .

Thirdly,   and   say things like: con-
cept x or y is the disjunction of concept x and concept y. And in doing so you might
think that   and   are committed to there
being some concepts that have the special status of being ‘disjunctive’ concepts.
Not so. I defined the concept x or y to be a concept for which the following is
knowable a priori: any given object is [insert concept x or y here] if and only if
the object is [insert concept x here] or the object is [insert concept y here]. And,
according to this definition, all concepts are disjunctions of some other concepts.
The concept is negatively charged, for example, is a disjunction of the concept is neg-
atively charged and in Australia and the concept is negatively charged and not in Australia.
As a result, my argument side steps the tricky issue of what it even means for a
property or concept to be disjunctive in any more substantial sense (Antony 1999;
Clapp 2001; Shoemaker 2007; Audi 2013)—as it does the tricky issue of what it
even means for a property or concept to be multiply realizable in any substantial
sense (Shapiro 2000; Polger 2008).

3 Defending Individuation by Necessary Coexemplification

I’ve suggested that Antony’s correlation argument is not as dialectically strong as
it might be. This is because many dualists will be unsure about one of its premises,
namely that properties are individuated by necessary coexempfication ( 
). For example, it looks like Shoemaker is committed to   

7



being true (McLaughlin 2009, §2), but Marras (1993, 291) and Haug (2011a) and
probably Fodor (1997) are committed to it being false. In the meta-ethics and
meta-normativity literatures, Streumer (2008, §4) reports that most dualists deny
  . This section therefore aims to lend force to the argument by
showing how    is supported on independent grounds.

One quick, independent way to motivate    would be to point out
that    follows from the Lewisian ontology of properties. Consider a
case in which, necessarily, any given individual exemplifies property A if and only
if it exemplifies B. It follows that exactly the same possible individuals exemplify
A as exemplify property B. But Lewis holds that properties just are these classes
of possible individuals. So A is the very same property as B. In short,  
: if property A and property B are necessarily coexemplified then A is the
very same property as B.

But this quick argument for    doesn’t go very far. My experience
is that dualists who are suspicious of    are equally suspicious of the
Lewisian ontology of properties. How then to lend force to   ? My
strategy for lending credence to    will be as follows. I will assume
the following metaphysical thesis about the nature of singular causation:

( ) Event C is a cause of event E if and only if event C makes
a difference to event E.

Then on the basis of  , section 4 will argue for

( ) Whenever any properties (for example P and Q ) are neces-
sarily coexemplified, then arbitrary event π (associated with P) and arbitrary
event ψ (associated with Q ) will have the same causes and effects.

But I will then assume

( ) If this arbitrary pair of events π and ψ have the same causes
and effects, then the associated properties P and Q have the same causal
powers.

Putting this together: whenever any properties (for example A and B) are neces-
sarily coexemplified, then they have exactly the same causal powers. Next I will
assume a principle of property individuation that is a putative alternative to 
 :

(  ) Whenever any property P and any property Q have exactly
the same causal powers then P is the very same property as Q.6

Put differently, no two properties have exactly the same causal powers. So for
property A and property B above, it follows that A is the same property as B. In

6See Armstrong (1978, §16.1), Shoemaker (1980), Swoyer (1982) and Fodor (1987, 44-45).

8



Monism
follows from follows from follows from

(I) 
(II)  

(III*)  (III)  
(IV)  

  
(V)   
(VI)  

(VII*)   (VII)  

section 2 section 3 section 4 and 5

other words,    follows: any necessarily coexemplified properties (A
and B) are the very same property. This is how I intend to lend credence to 
 , a central premise in the correlation argument.

Therefore the overall shape of the paper is as follows. Section 2 showed how
monism follows from five assumptions: (I) , (II)  , (III*) -
, (IV)  , and   . But, section 3 showed that
   in turn follows from three further assumptions: (V)   -
, (VI)  , and (VII*)  . And section 4 will show
that (VII*)   in turn follows from (VII)  . And
indeed section 5 will show that (III*)  in turn follows from (III)  -
. Thus, this paper shows how monism follows from seven assumptions in
total I–VII. (This information is summarized in table 1.) Again, my aim is not to
compel dualists to become monists, but to urge dualists to clarify their commit-
ments, more on which in section 6.

4 Mirrored Causes

In this section, I will assume  : singular causation is difference
making. In fact, I will consider two specific versions of  : singu-
lar causation is probability-raising (version one); and singular causation is coun-
terfactual dependence (version two). My task will be to show that each of these
versions of   issues in  : whenever properties
(for example P and Q ) are necessarily coexemplified, then the associated events π
and ψ have the same causes and effects.

A warning: this probability raising assumption and this counterfactual depen-
dence assumption are very crude assumptions about the metaphysics of singular

9



causation. These assumptions suffer from well-known objections, in particular
cases of causal pre-emption and over-determination; and the probability raising
assumption also suffers from confounding via common causes (Collins, Hall, and
Paul 2004). Nevertheless, the argument in this section can be modified to appeal
to much more sophisticated versions of  , I contend. So you
should think of the argument in this section as an idealized model, which can be
fruitfully used to build a more sophisticated argument—depending on the precise
nature of the more sophisticated version of   that you person-
ally endorse.7

Another warning: the argument in this section will talk in the abstract about
a property P and a property Q. To fix ideas more concretely, it might help to
imagine that P is the property is alkaline, and that Q is the property contains hydroxide
or ammonium ions. Of course, if you doubt that one or both of these properties exist,
then you will need to choose a different example with which to make the argument
in this section more concrete. It doesn’t matter what example you choose: the
argument of this section at no point appeals to any intuitions about any concrete
cases. It proceeds on an entirely general and abstract level.

With this caveat in mind, take any given object that exemplifies any given prop-
erty P at any given time. For example, tomorrow this liquid will be alkaline. Let
π be the associated event, the event of the liquid being alkaline tomorrow for ex-
ample. Then consider any property Q that is necessarily coexemplified with P.
Take for example the property of containing hydroxide or ammonium ions. And
consider the associated event ψ, for example the event of this liquid containing
hydroxide or ammonium ions tomorrow. In short: we will be considering simul-
taneous events π and ψ, which are respectively associated with the exemplification
of P (being alkaline for example) and of Q (containing hydroxide or ammonium
ions for example) in the liquid tomorrow.

The Probability-Raising Principle. Let’s start by assuming that singular causation
is probability raising. That is to say: event C was a cause of event E just in case
C preceded E, and C increased the probability of E. And to say that C increased
the probability of E is just to say that P(E|C) > P(E|C).8 This is mathematically

7For example, you might endorse a more sophisticated way of treating causation as difference
making. Something similar to what Halpern and Pearl (2005) and Halpern and Hitchcock (2015)
propose: event C was a cause of event E if and only if there is some unspecified event B such
that (a) BC > E; and (b) BC > E; where (c) C preceded E; and where (d) B is such that there
is a chain of dependence from singular variable C to singular variable E which doesn’t go via
singular variable B. And one can, I contend, use the RT and CSO axioms about counterfactual
conditionals to establish that condition d for example will hold for C = π if and only if condition
d holds for C = ψ.

8I follow the mathematical convention of using overlines to denote the logical operation of
negation. And P(|) denotes conditional probability.

10



equivalent to: P(EC) > P(E)P(C).9

The task in hand is to use the probability-raising principle to demonstrate
 . Let’s start by observing that P(π) = P(ψ). Indeed one can
show that P(Eπ) = P(Eψ) for any given event E. These claims follow from the
wholly uncontroversial axiom of probability theory that impossible events have
probability zero.10 It follows that (a) P(Eπ) > P(E)P(π) is equivalent to (a*)
P(Eψ) > P(E)P(ψ); just substitute the relevant terms. But π and ψ have the same
temporal location so (b) π preceded E if and only if (b*) ψ preceded E. Putting this
together: a and b hold if and only if a* and b* hold. But the probability-raising
principle says that a and b is equivalent to the claim that π caused E; similarly a*
and b* is equivalent to the claim that ψ caused E. Therefore for any given event
E: π caused E if and only if ψ caused E. In other words, π and ψ have exactly the
same effects.

Similarly recall that P(π) = P(ψ) and P(Cπ) = P(Cψ) for any given event C.
It follows that (a) P(Cπ) > P(C)P(π) is equivalent to (a*) P(Cψ) > P(C)P(ψ);
just substitute the relevant terms. But π and ψ have the same temporal location
so (b) C preceded π if and only if (b*) C preceded ψ. Putting this together: a and b
hold if and only if a* and b* hold. But the probability-raising principle says that a
and b is equivalent to the claim that C caused π; similarly a* and b* is equivalent
to the claim that C caused ψ. Therefore for any given event C: π was caused by
C if and only if ψ was caused by C. In other words π and ψ have exactly the same
causes.

The overall conclusion is  : π and ψ have exactly the same
causes and effects. The liquid’s being alkaline tomorrow has exactly the same
causes and effects as the liquid’s containing hydroxide or ammonium ions tomor-
row, to take just one concrete example (although recall my caveat at the beginning

9I follow the mathematical convention of using two conjoined letters such as EC to represent
the logical operation of conjunction.

10By definition π and ψ necessarily co-occur. So πψ is impossible, as is ψπ. So Zπψ is impossible,
as is Zψπ; for any given state of affairs Z. But it’s an axiom of probability theory that impossible
states of affairs have probability zero. So it’s a theorem that P(Zπψ) = 0 and that P(Zψπ) = 0.
But it is also a theorem of probability theory that

P(Zπ) − P(Zπψ) = P(Zπψ) = P(Zψ) − P(Zψπ)

By eliminating the zero terms we have

P(Zπ) = P(Zψ)

And letting Z be any tautology we also have

P(π) = P(ψ)

11



of the section about whether this concrete example is well-chosen).
The Counterfactual Dependence Principle. I will now argue for   via

an alternative route. I will assume an alternative version of  .
This version says that singular causation is counterfactual dependence. That is
to say: event C was a cause of event E just in case C preceded E, and if C had
been absent then E would also have been absent (Lewis 1986). (For brevity I will
express the latter counterfactual conditional formally as C > E.)

The task now is to use this counterfactual dependence principle to demonstrate
 . One can show that for any given event E: (a) π > E holds if
and only if (a*) ψ > E holds. This follows just from the CSO axiom governing
conditionals.11 But π and ψ have the same temporal location so (b) π preceded E
if and only if (b*) ψ preceded E. Putting this together: a and b hold if and only if
a* and b* hold. But the counterfactual dependence principle says that a and b is
equivalent to the claim that π caused E; similarly a* and b* is equivalent to the
claim that ψ caused E. Therefore for any given event E: π caused E if and only if
ψ caused E. In other words π and ψ have exactly the same effects.

One can also show however that for any given event C: (a) C > π holds if and
only if (a*) C > ψ holds. This follows from the RT axiom governing conditionals.12

But π and ψ have the same temporal location so (b) C preceded π if and only if (b*)
C preceded ψ. Putting this together: a and b hold if and only if a* and b* hold.
But the counterfactual dependence principle says that a and b is equivalent to the
claim that C caused π; similarly a* and b* is equivalent to the claim that C caused
ψ. Therefore for any given event C: π was caused by C if and only if ψ was caused
by C. In other words π and ψ have exactly the same causes.

Again the overall conclusion is  : π and ψ have exactly the

11Whenever one state of affairs such as ψ necessitates another such as π, then we have π > ψ;
and whenever π also necessitates ψ, then we also have ψ > π. It follows that if π > E then ψ > E,
for any given state of affairs E. This is because the material conditional ‘if π > ψ and ψ > π, and if
π > E, then ψ > E’ is just an instance of the CSO axiom for conditionals (Nute and Cross 2002,
10, 87-88). By a similar logic we have: if ψ > E then π > E. To grasp the CSO axiom consider
the conditional A > B: if Anil were to go the party then Beth would go to the party. And consider
B > A: if Beth were to go, then Anil would go. And A > C: if Anil were to go, Charlie would
go. And B > C: if Beth were to go, then Charlie would go. The CSO axiom is the material
conditional: if A > B and B > A, and if A > C, then B > C (Nute and Cross 2002, 10, 87–88).

12Whenever one state of affairs such as ψ necessitates another such as π then: if π were not to
obtain, then ψ would not either, no matter what other states of affairs also obtain. Therefore we
have Cπ > ψ; for any given state of affairs C. But the material conditional ‘if Cπ > ψ and C > π
then C > ψ’ is just an instance of the RT axiom for conditionals (Nute 1980, 18,23). It follows
that if C > π then C > ψ. And by a similar logic we have: if C > ψ then C > π. To grasp the
RT axiom consider the conditional A > B: if Anil were to go to the party, then Beth would go
to the party. And consider AB > C: if Anil and Beth were to go, then Charlie would go. And
consider A > C: if Anil were to go then Charlie would go. The RT axiom is the following material
conditional: if A > B and AB > C then A > C.

12



same causes and effects. The liquid’s being alkaline tomorrow has exactly the
same causes and effects as the liquid’s containing hydroxide or ammonium ions
tomorrow, to take just one concrete example. Note that to support this conclu-
sion, I didn’t appeal to anything as controversial as the Lewis–Stalnaker semantics
for conditionals. I merely appealed to the RT and CSO axioms. And these ax-
ioms are relatively uncontroversial; see Nute (1980, 18, 23, 29, 52, 63, 128–31)
and also Nute and Cross (2002, 87–88). For example, these axioms hold even on
Woodward’s interventionist semantics for conditionals (Galles and Pearl 1998).

Summary. I have assumed, very crudely, that singular causation is probability
raising or counterfactual dependence. On this basis I’ve argued for 
: whenever any properties (for example P and Q ) are necessarily coexem-
plified, then arbitrary event π (associated with P) and arbitrary event ψ (associated
with Q ) will have the same causes and effects. The liquid’s being alkaline tomor-
row has exactly the same causes and effects as the liquid’s containing hydroxide
or ammonium ions tomorrow, to take one concrete example. The same conclu-
sion also follows, I contend, for less crude versions of the assumption that singular
causation is  . (At this point it’s worth repeating the caveat at
the beginning of this section that this concrete example may not be well chosen,
and also to repeat that the aim of the present paper is not to establish monism
conclusively.)

5 The Parity Thesis and ‘The Property of’ Locutions

I now turn my attention from    to another controversial assump-
tion of the correlation argument, namely . My aim in this section will be to
lend some credence to  by showing that  follows from an extremely
pedestrian thesis ( ) about the semantics of ‘the property of ’ lo-
cutions. It is not entirely unreasonable to deny  , I acknowledge,
but section 6 will point out the costs of doing so.

Let’s begin by noticing that monists and dualists often use locutions of the form
‘the property of ’—for example ‘the property of being alkaline’ or ‘the property of
containing either hydroxide or ammonium ions’. But what does it mean to talk of
‘the property of being alkaline’ for example? Here’s a first attempt at an answer,
which I will improve on in a moment. ‘The property of being alkaline’ denotes the
property X that meets the following conditions: (1a) the exemplification of prop-
erty X in any given object necessitates that the object is alkaline; and (2a) any given
object being alkaline necessitates that the object exemplifies property X. (When
I say that a state of affairs ‘necessitates’ a second state of affairs, I mean that its
absolutely impossible for the first state of affairs to obtain without the second ob-
taining also.) Of course, there may be no property that meets these conditions,

13



in which case the locution ‘the property of being alkaline’ will fail to denote any-
thing. Or indeed, there may be two or more properties X 1 and X 2 that meet these
conditions. In which case, there must be some extra condition that functions as a
tie-breaker, as it were. For example, one might think that ‘the property of being
alkaline’ denotes the property out of X 1 and X 2 whose exemplification somehow
‘grounds’ the object being alkaline, in addition to necessitating that the object is
alkaline (Fine 1994). Taking all this on board, and generalizing away from the
specific concept of being alkaline, yields a more rigorous principle:

( ) ‘the property of being [insert any concept a here]’ suc-
ceeds in denoting some property if and only if there is at least one property
that meets the following conditions: (1) the exemplification of property X in
any given object necessitates that the object is [insert concept a]; and (2) any
given object being [insert concept a] necessitates that the object exemplifies
property X.

In short,   is a very straightforward story about the conditions
under which ‘the property of ’ locutions succeed in denoting something (and a story
that narrows down what these locutions denote, when they do denote). And, as
far as I’m aware, the philosophical literature has not yet proposed any substantive
alternatives to  .

This puts us in a position to argue for . Let’s suppose just for the mo-
ment that ‘the property of being alkaline’ succeeds in denoting a property. 
 then says that there is at least one property that meets both conditions
(1a) and (2a) above. But let’s also suppose just for the moment that, necessarily,
any given object is alkaline if and only if it contains hydroxide or ammonium ions.
So condition (1a) above entails that (1b) the exemplification of property X in any
given object necessitates that the object contains hydroxide or ammonium ions;
and condition (2a) above entails (2b) that any given object containing hydroxide
or ammonium ions necessitates that the object exemplifies property X. This is
because necessitation is a transitive relation. So, since conditions (1a) and (2a)
above entail that conditions (1b) and (2b) are also met, and since there is at least
one property that meets conditions (1a) and (2a), there is at least one property
that meets conditions (1b) and (2b). But, since there is at least one property that
meets conditions (1b) and (2b),   says that the locution ‘the prop-
erty of containing hydroxide or ammonium ions’ succeeds in denoting a property.
Let’s discharge our assumption that the locution ‘the property of being alkaline’
succeeds in denoting some property. This yields: if the ‘the property of being
alkaline’ denotes some property, then ‘the property of containing hydroxide or
ammonium ions’ denotes some property too. Let’s also discharge our assumption
that, necessarily, any given object is alkaline if and only if it contains hydroxide or
ammonium ions. That is to say, let’s discharge our assumption that the concept
is alkaline is necessarily co-applicable with the concept contains hydroxide or ammonium

14



ions. Discharging this assumption, and expressing the conclusion in its most gen-
eral form, yields: () if a concept denotes a property, then any necessarily
co-applicable concept itself denotes some property.

To pre-empt some objections, it is of the utmost importance to note the fol-
lowing point:   is consistent with there being two properties that
satisfy conditions (1a) and (2a), and with there being two properties that satisfy
conditions (1b) and (2b). In this case, the tie-breaker condition springs into ac-
tion to make sure that ‘the property of being alkaline’ denotes a single property,
and ‘the property of containing hydroxide or ammonium ions’ denotes a single
property. And, for all   says, this tie-breaking condition may en-
sure that ‘the property of being alkaline’ denotes a property that is distinct from
‘the property of containing hydroxide or ammonium ions’. Of course, the corre-
lation argument will ultimately suggest that these two locutions each denote the
very same property. Nevertheless, in assuming  , I do not beg
the question here against the dualist by assuming at the outset that these two lo-
cutions each denote the same property. Put differently,   does
not trivialize the correlation argument: it does not make the other premises in the
argument redundant.13 (This same goes for , which is just a direct conse-
quence of  .)

In sum, to deny the  principle, the dualist would need to deny 
, my extremely pedestrian understanding of ‘the property of ’ locutions.
This is not a wholly unreasonable thing to deny, I’m happy to acknowledge. But
it doing so does come with some costs: for one thing, it trivializes the dispute
between correlation argument monists and dualists, more on which in a moment.

6 Conclusion: Dualists Need to Be Clearer

The debate between dualists and monists is one of the central debates in the phi-
losophy of mind, of biology and of the social sciences. My aim in this paper has
not been to provide decisive reasons for dualists to embrace monism—far from
it. The aim instead has been to move the intricate debate between monists and
dualists forward by sharply distinguishing the options available to the dualist.

Section 2 offered a new version of the correlation argument. Like Antony’s
version of this argument, my version has the virtue of being formulated as a de-
ductive rather than as a non-deductive argument; contrast the arguments hinted

13Note also that   does not entail that, if ‘the property of being alkaline’ denotes
a objectively important property, then ‘the property of containing hydroxide ions or ammonium
ions’ also denotes an objectively important property (section 2). And note also that  is a con-
ditional thesis. So the above argument does not show that ‘the property of containing hydroxide
ions or ammonium ions’ succeeds in denoting a property.

15



at in Bacon (1986), Kim (1992), or Clapp (2001). And it also has the virtue of
remaining neutral on the issue of the existence and objective importance of mul-
tiply realizable properties and disjunctive properties; contrast Antony (2008) and
Clapp (2001). What’s more, it has the virtue of making all its assumptions fully ex-
plicit. These assumptions are (I) , (II)  , (III*) , (IV)
 , as well as   . And sections 3 and 4 strength-
ened the dialectical force of the correlation argument by showing that one of its
controversial premises (  ) follows from some commonly accepted
premises, namely (V)   , (VI)  , and (VII) 
. And section 5 strengthened its dialectical force even further, by showing
that another controversial premise () follows from (III)  .
In short, sections 2–5 showed that dualists must reject at least one of the theses
I–VII.

For a minority of dualists, it is already clear which of theses I to VII they would
reject. Crane and Mellor (1990) reject , for example, and dualists attracted
by a hyper-intensional approach to metaphysics would probably reject -
  (Jenkins and Nolan 2012; Nolan 2014), I suspect. However, for the
majority of dualists it is entirely unclear which of I to VII they would reject.

Indeed, for each thesis from I to VII there are some dualists who are clearly
committed to that thesis. Firstly, I’ve already noted that most dualists endorse
, or at least seem to. Secondly,    is the centrepiece of most
dualists’ metaphysical picture; see Fodor (1987, 44-45) and Shoemaker (2007) for
example. Thirdly, Fodor (1974, 1997), Pereboom and Kornblith (1991), Block
(1997), Sawyer (2002), Haug (2011a, 2011b) and many other dualists are clearly
committed to   and to  and to  . In-
deed, they are happy to agree that there are an abundance of physical properties
for example, such as the property contains hydroxide or ammonium ions (Baker 1993,
81). Instead, these dualists insist that such properties are most of them objectively
unimportant; see section 2 and Antony (1999) for discussion.

Fourthly, many dualists are committed to  : singular cau-
sation is difference making. To see this, note that Le Pore and Loewer (1987),
Horgan (1989, 1997), Baker (1993), Bennett (2003) and List and Menzies (2009)
and many other dualists rely upon   in order to undermine the
exclusion principle at the heart of Kim’s infamous exclusion argument. Kim’s ex-
clusion principle says that it is rare for a single event at time t1 to have two distinct
but ‘complete’ causes at time t0. Therefore a patient’s experiencing a reduced
level of pain, for example, was not caused both by the event of her ingesting co-
diene and also by the event of her ingesting an analgesic. But, since the reduced
level of pain was caused by the event of her ingesting codiene, it was not caused
by the event of her ingesting an analgesic, Kim would argue. In order to reject

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Kim’s conclusion here (and the exclusion principle upon which it is based) the
above dualists appeal to  : the event of ingesting an analgesic
made a difference to her reduced levels of pain, and therefore was a cause of it.
Note that these dualists do not respond to Kim by claiming  ∗:
difference making is sufficient for causation, but only if certain extra conditions
hold. (Example extra conditions: ingests an analgesic is a objectively important prop-
erty; ingests an analgesic has the right sort of essence.) After all, if these dualists were
to rely only on  ∗, then they would have the extra burden of
showing that these extra conditions do in fact hold in the case of the analgesic.
And this would render their objection to Kim’s exclusion principle dialectically
ineffective, and indeed utterly question begging.

In sum, for the majority of dualists it is entirely unclear which thesis of I–VII
they would reject; and each for thesis from I–VII there are some dualists who are
clearly committed to that thesis. My hope is to push the debate forward by putting
pressure on dualists to be explicit about which I–VII they will reject and why. I
expect that doing so will uncover considerable heterogeneity amongst dualists.

7 The Costs of Dualism

I will finish up by reviewing the options available to the dualist. Dualists need to
deny at least one of the following seven assumptions.

(Assumption I ) The following is absolutely necessary: whenever any
object exemplifies precisely those properties given by any a-compatible total
conjunction of S-properties, then this object is [insert concept a here].

That is to say, concept a supervenes on the S-properties (on the physical proper-
ties for example). In my view, it is completely reasonable for monists and dualists
to disagree over . The disagreement here might stem from a compara-
tively trivial disagreement over what properties count as S-type properties (what
properties count as physical properties for example). Or the disagreement might
be more substantial. For example, a dualist might think that  only holds
when ‘absolutely necessary’ is replaced with ‘nomically necessary’, the necessity
associated with the laws of nature. If so, there are some generalizations that are
nomically necessary (lawful) but that are not absolutely necessary. In which case,
the dualist ought to say what exactly it is for a generalization to be nomically nec-
essary (lawful) in the first place. What’s more, she ought also to show that her
reasons for endorsing  as a claim about nomic necessity aren’t themselves
reasons for also endorsing  as a claim about absolute necessity. Either way,
the dualist should make her reasons for denying  explicit.

(Assumption II  ) For any two concepts x and y, there is a

17



concept x and y that is their conjunction, and a concept x or y that is their
disjunction, and there is a concept that is their negation, not x for example.

As I’ve already discussed in section 2, the prospects for rejecting  
are not good. Of course, if one restricts one’s attention to concepts that are hu-
manly graspable, then   will be false. But doing so risks turning
this debate into a debate about subjective human limitations, not a debate about
objective metaphysics (Clarke 2017).

(Assumption III  ) ‘the property of being [insert any concept
a here]’ succeeds in denoting some property if and only if there is at least
one property that meets the following conditions: (1) the exemplification of
property X in any given object necessitates that the object is [insert concept a
here]; and (2) any given object being [insert concept a here] necessitates that
the object exemplifies property X.

Section 5 has explained why   is much less controvertial than it
may first appear. Note also that   may be true for some uses of
‘the property of ’ locutions and false for other uses. Indeed, as far as monist users
are concerned,   makes sense of everything that monists say. So
the most charitable interpretation of such monists is that   is true
of monist uses of ‘the property of ’ locutions. Consider then a dualist who concedes
that   is true for monist uses of ‘the property of ’ locutions, but
who insists that it is false for dualist uses. To some extent, this is a reasonable move
for the dualist to make. But this move would radically reconfigure how the monist
versus dualist debate is typically framed: the debate becomes a semantic debate
in which the monist and dualist are talking past each other, by using ‘the property
of ’ locutions in different ways. Furthermore, this move burdens dualists with the
following challenge: how is the reference of ‘the property of ’ locutions determined
for dualist users, if not by  ? To the extent that dualists can’t give
a substantive answer to this question, dualism is a less explanatory theory, one that
creates a mystery where there need be no mystery.

(Assumption IV  ) Take any class of concepts, all of which
denote properties of type S. And take any concept that is formed purely out
of taking conjunctions and disjunctions and negations of the concepts in this
class. Then if this latter concept denotes a property, this property is itself of
type S.

For example, if a property is denoted by the concept contains hydroxide or contains
ammonium ions, then this property is a physical property, because the concept con-
tains hydroxide ions denotes a physical property and the concept contains ammonium
ions denotes a physical property. So far as I can see,   is the
only way of justifying or explaining—in a systematic and principled manner—why
paradigm physical properties deserve to be counted as physical. Take for example

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the property has kinetic energy of less than ten joules or the property is vidocin or ibuprofen or
codiene; and see Clarke (2017) for discussion. Nevertheless, it’s completely reason-
able for dualists to disagree with  , I think. But such dualists
should acknowledge that, in doing so, they are jettisoning a more systematic and
principled approach to classifying properties as physical in favour of a more in-
tuitive and less systematic approach. At any rate, denying  
makes the disagreement between dualists and ‘correlation argument monists’ a
comparatively superficial disagreement over what properties to label as physical
properties, for example. So, once this has been revealed to be the fundamental
point of disagreement between a monist and a dualist, then there is no point in
them pursuing this disagreement further, I urge. If so, why has this point (that
the disagreement is a trivial one over  ) not been recognized
sooner? one might ask. Here the blame lies more with dualists, I think, because
  has been a well-advertised feature of many monists’ philoso-
phies (Kim 1998, 82).14

(Assumption V   ) Whenever property P and property Q have
exactly the same causal powers then P is the very same property as Q.

To some extent, it is reasonable for monists and dualists to disagree over  
, I think. But denying    introduces mysteries where there
need be none, namely about how properties are individuated. And it also creates
some corresponding epistemological problems for the dualist. Absent some alter-
native principle about how to individuate properties, how come you are confident
that mental properties are not physical properties, for example, or that normative
properties are not natural properties? Blind faith alone? Or because you are un-
wittingly assuming that no two concepts denote the same property? If the latter,
then this trivializes the debate: questions about property identity are no longer
questions about objective metaphysics, but are instead questions about human
concepts (Clarke 2017).

(Assumption VI  ) If an arbitrary pair of events π and ψ have
the same causes and effects, then the associated properties P and Q have the
same causal powers.

I myself can see no motivation for denying this assumption.

(Assumption VII  ) Event C is a cause of event E if and
only if C makes a difference to E.

Here difference making can be defined crudely in terms of probability raising or
counterfactual dependence, or it can be given a more sophisticated definition; see
Halpern and Hitchcock (2015) for example. At any rate, it’s reasonable for du-
alists to deny  , I think, for all sensible definitions of difference

14I suspect that McLaughlin (2009, §2) makes this criticism of Shoemaker’s version of dualism.

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making. But doing so introduces mysteries where there need be none, namely
about the metaphysics of singular causation. And it creates some corresponding
epistemological problems for the dualist: why do you think that mental events can
cause other events? how do scientists come to know what causes what, other than
by examining difference making? Causal inference is difficult enough without
philosophers introducing additional and unwelcome obstacles.

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