Causation sans Time
Abstract
Is time necessary for causation? We argue that, given a counterfactual theory of causation, it is
not. We defend this claim by considering cases of counterfactual dependence in quantum
mechanics. These cases involve laws of nature that govern entanglement. These laws make
possible the evaluation of causal counterfactuals between space-like separated entangled
particles. There is, for the proponent of a counterfactual theory of causation, a possible world in
which causation but not time exists that can be reached by ‘stripping out’ time from the actual
world, leaving (some) quantum mechanical laws intact.
1. Introduction
We typically think of causes and effects as events that are temporally related. Though
relatively little has been said to defend the claim, it seems to be tacitly assumed that it is an
essential feature of causation that its relata are related in this way.1 In this paper we argue that,
given a particular conception of causation, causation without time is possible. We focus on a
counterfactual theory of causation according to which counterfactual dependence between x and
y is sufficient for y’s causal dependence on x.
The counterfactual theory of causation, popularised by Lewis (1973a), has been recently
defended by a number of authors. Frisch (2010) and Coady (2004) defend the semantics of the
counterfactual theory against various objections, while Menzies (1996) and Hall (2004) argue that
there is at least one meaningful concept of causation according to which counterfactual
1
This is particularly evident for process theories of causation, according to which causation involves the intersection
of world-lines in space-time (see e.g. Salmon (1994) and Dowe (1992)). But it also appears apt for counterfactual
theories of causation (see Lewis (1973a, 1973b, 1979)).
1
dependence is sufficient for causal dependence.2 Schaffer (2000, 2005), though critical of the
counterfactual theory, is ultimately sympathetic to the broad picture, attempting to parlay his
contrastive theory of causation into a counterfactual framework. Finally, Woodward (2003)
defends a version of the counterfactual approach, focusing on certain ‘interventionist’
counterfactuals; counterfactuals concerning what would occur for some y were one to intervene
on some x.
Tallant (2008) has recently argued that if one adopts a counterfactual theory of causation,
then possibly, there is causation without time. We think that Tallant’s argument fails but that
there is a sound argument in the neighbourhood. Our aim is to elucidate that argument. This is
important, first, given recent developments in physics. In the quest for a grand unified theory
that reconciles general relativity with quantum theory to produce a theory of quantum gravity it
has been argued that we must deny the existence of time (see, e.g. Barbour (1999) and Rovelli
(2007)). The suggestion that time does not exist is a radical proposal. As Healey (2002) has urged,
any theory that denies the existence of time is faced with a substantial reconstruction project
whereby the appearances as-of temporality are systematically recovered. If it is possible that
causation exists without time then the reconstruction project looks promising since it seems
more likely that we can recover these appearances if we can recover causation.
Second, the argument is important because it furnishes additional reasons to accept, or reject,
the counterfactual theory of causation. Those who think that causation in the absence of time is
impossible will have (perhaps further) reason to reject the counterfacutal theory of causation,
while those who are more sympathetic to such a proposal—such as those interested in a
reconstruction project—will have reason to embrace a counterfactual theory of causation. We
are more tempted by the latter than the former, but will not seek to defend that claim in detail
2
Though this is not the only causal concept in use. Another concept lines up, roughly, with the process accounts of
causation of Salmon (1994) and Dowe (1992).
2
here. Our primary focus lies with revealing the conceptual relationship between causation and
time.
2. Definitions and Assumptions
2.1
Assumptions
We begin with two assumptions of the paper and then some terminology. First: the
assumptions. We assume that time is not, of necessity, characterised by temporal passage of the
kind posited by A-theorists. Rather, B-relations—relations of earlier than, later than and
simultaneous with—are both necessary and sufficient for the existence of temporality. Further,
we assume that either B-relations are irreducible primitives (i.e. they are external relations) or
that, if they are reducible, their reductive base is not the set of causal relations. If B-relations
supervene on, or are determined by, the set of causal relations in a world, then there can be no
time without causation and the debate is foreclosed.
Moreover, as previously noted, we assume that counterfactual dependence is sufficient for
causation. This condition is typically stated in terms of events: ‘if event e had not occurred, event
c would not have occurred’ entails the causal statement ‘e caused c’. We will return to the notion
of an ‘event’ in a moment. First, though, it is important to say a bit more about the sufficiency of
counterfactual dependence for causation before pressing on.
One of the central difficulties with the sufficiency condition is that – without further
clarification – it is too broad. As Kim argues, there are counterfactuals that, while true, should
not imply causal dependence. The counterfactual theory must therefore impose some constraints
on the space of counterfactuals casting a net around only the causal ones. Kim offers the
following examples:
3
(1) If yesterday had not been Monday, today would not be Tuesday.
(2) If George had not been born in 1950, he would not have reached the age of 21 in 1971.
(3) If I had not written “r” twice in succession, I would not have written “Larry”.
(4) If I had not turned the knob, I would not have opened the window.
(5) If my sister had not given birth at t, I would not have become an uncle at t.
In response, Lewis (1986) argues that causal counterfactuals are counterfactuals that hold
between distinct events where events have the following features: they are (i) matters of contingent
fact regarding properties localised to regions, (ii) predominantly intrinsic, (iii) capable of standing
in relations of implication (e.g. the proposition <event x occurs> might entail <event y occurs>)
(iv) non-disjunctive and (v) stand in parthood relations to other events: both directly and
indirectly, by being parts of the regions that other events occupy. Two events, x and y, are
distinct when (a) x and y do not stand in relations of implication and (b) x is not part of y or vice
versa.3
The restriction to events that are distinct takes care of Kim’s examples (see Lewis (1986) for
details). We deploy the same restriction to mark out causal from non-causal counterfactuals. We
recognise, however, that there are ongoing concerns regarding the asymmetry of counterfactuals
3
One might press us here on the difference between events and things. The main ontological difference between an
event and a thing, it could be argued, is that a thing has no temporal extension: it is wholly present at every moment
at which it exists. Events, by contrast, do have temporal extension: they are not wholly present at every moment at
which they exist. Below we liberalise the notion of an event to allow for purely spatial instances. This might seem to
collapse the ontological distinction between things and events: if events, like things, can lack temporal extension,
then what is the difference, really? It is not obvious to us, however, that the ontological difference between things
and events is to be posed in such terms, for two reasons. First, we accept perdurantism and so believe that things,
like events, fail to be wholly present at each moment at which they exist. Second, even if we give up perdurantism, it
seems conceptually possible for an event to be wholly present and to thus endure like a thing. Either way, things and
events are not to be differentiated in terms of their temporal features.
4
and there may be further recalcitrant cases along the lines identified by Kim. 4 Since we are
interested in revealing the relationship between causation and time given a counterfactual theory
of causation rather than in defending that theory, we won’t seek to defend the sufficiency claim. 5
We therefore assume a standard Lewis-Stalnaker semantics for causal counterfactuals in terms of
a closeness metric over possible worlds. In particular, we will be primarily interested in matching
worlds based on similarity in the laws of nature.
A further worry: since we hold that only counterfactual dependencies that obtain between
distinct events are thereby causal dependencies, we need a notion of an event that is not
essentially temporal. Lewis (1986, pp. 243–244) tells us that events are localised to regions, but
he clearly has in mind spatiotemporal regions. The mention of temporality is, however, nonessential since actual events can have counterparts in worlds without time. Consider a plane, P,
of simultaneity through a four-dimensional space-time. Generally accepted views about modal
separability and recombination suggest there is a world, w, composed of nothing but an intrinsic
duplicate of P. In w there exists only a spatial dimension. But any instantaneous event (or event
part) that occurred on P will have an intrinsic duplicate in w. That will be an event that is located
at a region, but not a spatiotemporal region.
2.2
4
Definitions
As pointed out to us by an anonymous referee, there are also doubts that (i) Lewis’ semantics for counterfactuals is
similar to the counterfactuals of causation and (ii) that the evaluation of counterfactuals is not highly context
dependent. See, for discussion, the collected papers in Collins, Hall, and Paul (2004).
5
Even if Lewis’ counterfactual picture fails to deliver the sufficiency of counterfactual dependence for causation,
there are other versions of this general strategy that may fare better, such as Woodward’s (2003) more recent
interventionist account.
5
We move now to some terminology. We will say that S is a slice of a world w iff (i) S is a threedimensional configuration,6 where each dimension is a spatial one; (ii) every property, event, or
object located on S bears some spatial relation to every other property, event, or object located
on S, and no property, event, or object located on S bears any temporal relation to any property,
event, or object located on S and (iii) there is no property, P, event, E, or object, O, that bears a
spatial but not temporal relation to some property, event or object located on S, such that any of
P, E, or O fails to be on S. In a world with B-relations slices are the relata of those B-relations.
We assume that there is a possible world that contains slices but does not contain B-relations,
and thus that there are possible worlds without time.
We also need the notion of laws that, as Skow (2007) puts it, “govern the evolution of the
world”. The term ‘govern’ is meant to be agnostic between Humean and non-Humean laws of
nature and the term ‘evolution’ is not meant to impute any essentially temporal aspect to the
laws. The phrase ‘govern the evolution of the world’ indicates that these are laws from which
one can determine what is going on at one region of the universe based on what is going on at
some distinct region. Laws of this kind underwrite projectability. Call these world-evolving laws. Skow
(2007, p. 237) offers a definition of such laws in terms of spatiotemporal regions. We want to
focus primarily on slices and spatial regions and so have adapted the definitions accordingly. This
might be seen to involve an untoward separation of space and time. However, everything we say
can be translated back into spatiotemporal regions if need be. Our reason for separating space
and time is primarily to make the discussion cleaner. Here is our definition of a world-evolving
law:
A set of laws, L1, L2, L3 ... Ln, are laws that govern the evolution of the world iff the Ln
6
These definitions could be generalised to include four-dimensional spatial configurations We focus on three-
dimensional configurations, since this lines up more intuitively with the notion of a slice: i.e. a three-dimensional
subsection of an n-dimensional universe.
6
together with a complete description of the local matters of fact that obtain at a spatial
region, R, yields complete information about (or assigns probabilities to complete
descriptions of) the local matters of fact that obtain at some region R* such that R ≠ R*.
Notice that the limiting case of a spatial region is a slice. Notice also that the Ln can be
deterministic or indeterministic. In addition, such laws can be time-evolving. We define a timeevolving set of the Ln as:
A set of laws, L1, L2, L3 ... Ln, are time-evolving iff they govern the evolution of the
world across time; i.e. the Ln together with a complete description of local matters of fact
at a slice S, yields complete information about (or assigns probabilities to complete
descriptions of) the local matters of fact that obtain at slice S* such that S ≠ S* and
where S and S* are B-related.
Examples of time-evolving laws are: Newton’s laws of motion, the second law of
thermodynamics, Maxwell’s equations of electro-magnetism and the time-dependent Schrödinger
wave-equation. These laws describe the dynamics of a system over time. For Skow, the only
world-evolving laws are time-evolving laws. This will become important later on. For now, we
want to leave it open whether there are any world-evolving laws that are not essentially timeevolving laws.
2. Causation without Time
According to Tallant (2008) we can show that causation without time is possible by simply
“removing” the B-relations from one world—call it wB—and leaving the rest of the world intact.
Since the stripped-down world—call it wS–will support the same counterfactuals as wB it will a
7
fortiori support causation. But since wS has, by stipulation, no B-relations, it has no time. Hence
wS is a world that lacks time but contains causation. Here, more precisely, is our reconstruction
of Tallant’s argument:
1. B-relations are necessary and sufficient for the existence of temporal relations.
2. wB contains B-relations
3. wB contains temporal relations (from 1 and 2)
4. wB contains causal relations (stipulation)
5. wS is just like wB except that wS does not contain B-relations
6. wS contains no temporal relations (from 5)
7. wS supports the same counterfactuals as wB (from 5)
8. wS contains the same causal relations as wB (from 7)
9. Therefore wS contains causal relations but not temporal relations.
In what respects is wS just like wB except that wB contains B-relations while wS does not? At a
minimum, we assume (i) for every slice in wB there is a slice in wS that is an intrinsic duplicate of
that slice and (ii) there is no slice in wS that is not an intrinsic duplicate of some slice in wB, and
(iii) there is a 1:1 mapping of the slices. If (i), (ii) and (iii) hold then wS is what we will call a
minimal intrinsic slice duplicate: it contains an intrinsic duplicate of every slice in wB, and contains no
more slices than that. There are two further assumptions one might make to secure the closeness
of wS and wB. First, one might assume that the 1:1 mapping function F between slices of wB and
wS is order-preserving. That is, the function maps each slice into its counterpart in a way that
preserves the ordering of slices between wB and wS. Second, one could assume that wB and wS
have the same laws. In what follows, we accept the first assumption but not the second. We
remain agnostic on the second because, as we shall see, whether or not w S is a world with causal
8
counterfactuals depends on whether or not wB and wS have the same laws. Assuming that they
do from the outset gives the game away.
The central problem with Tallant’s argument is premise (7). The problem takes the form of a
dilemma. Do wB and wS share the same laws of nature? Suppose the answer is ‘yes’. Then the
existence of B-relations, and thus of time, makes no physical difference to the universe. That is
hard to swallow. It is not that the presence/absence of time must make a physical difference to
the universe by appearing in the laws of nature.7 Nor are we suggesting that any difference in the
distribution of matters of fact in a world necessitates a difference in the laws in that world.
Rather, recall that wB and wS are intrinsic slice duplicates. The intrinsic matters of fact in wS are
the same as in wB. So the presence/absence of time can be making no difference to the intrinsic
character of the slices, and no difference to the laws. So the question is, where else could the
difference show up? Not in any experimental finding, since the slices are intrinsically the same.
But if there is no empirical difference between the two worlds, then it is difficult to see what
grounds we have for believing that time is physically relevant.
Suppose that the answer to the above question is ‘no’: wB and wS do not have the same laws
of nature. Then (7) is false. For on Lewis’ standard picture of counterfactuals, the laws of nature
underpin counterfactual dependence. If the laws of nature are changed, then the counterfactual
dependencies vary accordingly. Thus, if the move to wS from wB involves a change in the laws of
nature, then wS does not support the same counterfactuals as wB.
3. Time-Evolving Laws
There is a weakening of (7) that avoids our argument:
(7*) wS supports counterfactual dependence (from 5)
7
We are indebted to an anonymous referee for pressing us on this issue.
9
Which requires a corresponding weakening of (8*):
(8*) wS contains causation (from 7*)
Something like (7*) is considered by Tallant (see e.g. pp. 121–122)). (7*) avoids the second horn
of the dilemma: Even if wS does not support the same counterfactuals as wB it supports some
counterfactuals.
For this weakening to work the move from (5) to (8*) must be secured. It must be shown that
because wS is just like wB except that wS does not contain B-relations, wS is capable of supporting
causal counterfactuals. We think the weakening is a good one and the resulting argument is
sound. In what follows, we defend this claim. To evaluate the move from (5) to (7*), we need to
consider what difference the removal of B-relations makes to the laws of nature. Since we are
assuming that B-relations are both necessary and sufficient for the existence of time, the removal
of B-relations would, at a minimum, result in the loss of what we have called time-evolving laws.
At wS there would be no set of laws L1, L2, L3 ... Ln, such that the Ln together with complete
information about the world at a slice S, yields complete information about (or assigns
probabilities to complete descriptions of) a slice S* such that S ≠ S* and where S and S* are Brelated. This is simply because there are no B-related slices at wS and so a fortiori no laws
involving them.
Arguably, the loss of time-evolving laws does not show that there is no counterfactual
dependence in wS. If there are world-evolving laws that are not temporal, then although wS might
lack time-evolving laws, it might possess evolving laws of this more general kind. If there are at
least some laws that govern in this more general way then there is likely to be counterfactual
dependence. If information about a slice, S1, plus the evolving laws Ln yields complete
information about (or assign probabilities to complete descriptions of) a distinct slice S2, then a
10
range of counterfactuals involving S1 and S2 are likely to be true, and this might be so even if
there are no B-relations. For instance, some counterfactual of the form: if x had not existed at S1,
then y would not have existed at S2 is likely to be true, since if there are world-evolving laws of
this kind then changes to S1 are likely to make a difference to S2.
This reveals a strategy for defending the move from (5) to (7*). If one could show (i) that
there are some world-evolving laws that are not also time-evolving laws and (ii) that wS is likely
to have such laws, then that would be a reason to think that (iii) wS is a world with counterfactual
dependence but no time. If we could then show (iv) some counterfactual dependencies in w S are
causal dependencies because the relata of the dependency relation are distinct events, we would,
in addition, have secured the move from (7*) to (8*).
Let’s start with (i). For this, we need a world-evolving law that is not temporal. One place to
turn is to the existence of laws that govern the evolution of the world across space. Call these
space-evolving laws. If there are such laws, then it is likely that they would survive the process of
stripping B-relations from a world and thus likely that certain counterfactual dependencies would
remain.
Skow (2007, pp. 243–249) is doubtful that such laws exist. He considers a handful of
candidates, but rejects them all as genuine cases of space-evolving laws. For Skow all worldevolving laws are time-evolving laws. If Skow is correct, then the move from (5) to (8*) can be
blocked by arguing that the existence of world-evolving laws is both necessary and sufficient for
counterfactual dependence. Then the move from wB to wS, via the removal of B-relations would
imply the loss of all world-evolving laws and wS would not be sufficiently similar to wB to
support counterfactuals and so a fortiori it would not be similar enough to support causal
counterfactuals.
We do not think that Skow is correct. There is at least one actual example in which spaceevolving laws play a role, a case that Skow does not consider. Take a simple quantum system, Q,
describable by the time-independent Schrödinger wave-equation:
11
2
ℏ 𝑚2 𝑚(𝑚)
2𝑚
𝑚𝑚2
+ (𝑚 − 𝑚(𝑚))𝑚(𝑚) = 0
(A)
Suppose that Q possesses the following features:
(i)
Q is a two-particle system, consisting of the particles P1 and P2
(ii)
P1 and P2 are entangled
(iii)
P1 and P2 possess a single quantum property: spin
(iv)
There are two possible combinations of spin properties for P1 and P2:
a. {|↑>, |↓>} (i.e. P1 has spin up and P2 has spin down)
b. {|↓>, |↑>} (i.e. P1 has spin down and P2 has spin up)
P1 and P2 are in motion and move at a constant velocity relative to each other. Q is formed when
the two particles meet at time t1 and interact, becoming entangled. After t1, P1 and P2 move apart
until they are separated by a spatial distance, D, (D’s value doesn’t really matter, so long as it is
>0). At some time, t2, such that t1 > t2, P1 hits a detector and its spin is measured. At t2, either
P1’s spin is up or it is down. We can model the situation as follows:
t
Detector
t2
P1
P2
t1
x
12
If, at t2, P1’s spin is up then because P1 and P2 are entangled it follows that, at t2, P2’s spin will
be down. Similarly, if P1’s spin is down then it follows that P2’s spin is up. That is, at t2 the
following two conditionals are true: |P1↑> → |P2↓> and |P1↓> → |P2↑>. Accordingly, at t2, if
we take all of the information about P1, including that it has been measured, that it is entangled
with P2 and that its spin is (say) up, then that information fully determines information about P2
at that time. The determination of P2 by P1 is happening across space; it is not the case that P1
determines P2 via some time-evolving law. Of course, the relevant laws can also be used to
project the dynamics of the system, via the time-dependent Schrödinger wave-equation (note the
presence of the time variable t in the right-hand side of the equation, which is not in the timeindependent wave-equation):
2
−
ℏ 𝑚2 𝑚
2𝑚 𝑚𝑚2
+ 𝑚(𝑚)ᴪ = 𝑚ℏ
𝑚𝑚
𝑚𝑚
(B)
However, it remains the case that there is a decidedly spatial aspect to the way in which the
laws govern the relationship between P1 and P2. If we evolve the world across space from P1 to
P2, then we can determine information about P2 based on information about P1. Although Q is a
toy example, the quantum mechanical laws that govern a simple quantum system of this kind are
the ones that govern the actual universe. We therefore have substantial empirical evidence
(namely, the evidence in favour of quantum theory) that cases relevantly analogous to this one
actually occur, and so we have evidence that at least some of the laws that govern our universe
are space-evolving.
Of course, matters are complicated by the manner in which the case was set up. We had to
assume that a measurement is carried out on P1, and that P1 and P2 interact at time t1 becoming
entangled as a result. Measurements are, plausibly, processes in time, as are the interactions that
produce entanglement. So the case has both a temporal and a spatial aspect. But the temporal
13
aspects are not essential to the determination of P2’s spin property across space. Once the wavepacket has been collapsed, this determination becomes possible.
Now, imagine a simplified world, wD, in which (i) there is a space-evolving law governed by
the time-independent Schrödinger wave-equation; (ii) there are just two slices S1 and S2 in w (iii)
there are no B-relations between S1 and S2 or, indeed, in the world in general and (iii) P1 and P2
exist at S1 and are such that both particles are entangled and have determinate quantum
properties. In such a world, information about P1 and the quantum system in which it is
embedded provides full information about (or assign probabilities to descriptions of) P2 at S1.
Moreover, in wD, a range of counterfactual conditionals are true. If in wD |P1↑>, then: ~|P1↑>
□→ ~|P2↓> will be true. That’s because the closest possible worlds to wD in which ~|P1↑>
holds, will be a world with the same laws and, in particular, the same space-evolving law as
specified in (i). Hence, in that world, P1’s spin determines, across space, the spin of P2 and so if
P1 is not spin up, then P2’s spin will not be down. Similarly, if |P1↓> then it follows that: ~|P1↓>
□→ ~|P2↑>. Note that the laws in this case are symmetrical. We can just as easily determine
information about P1 from P2. As such, if |P2↓>, then ~|P2↓> □→ ~|P1↑> and if |P2↑>, then
~|P2↑> □→ ~|P1↓>.
It remains to be shown that (i) we can get to a world of the kind just discussed by stripping
out the B-relations from some other world, thereby securing the inference from (5) to (7*) and
(ii) these counterfactuals are causal counterfactuals, thereby securing the inference from (7*) to
(8*). In fact, this can be shown with the actual world. First, take the actual world (call it wB) and
foliate it into slices S1, S2, S3 ... Sn connected by B-relations.8 Take a function, F, that maps the Sn
into the slices S*1, S*2, S*3 ... S*n in some other world, wS, to create a minimal intrinsic slice
duplicate in which the ordering of slices has been preserved but in which there are no Brelations.
8
That is not to say that there is a unique foliation on the actual world, just that there is some foliation that can be
done, which is compatible with the special and general theories of relativity.
14
We assume that wS will have different laws of nature to the actual world since there won’t be
any time-evolving laws. The space-evolving laws, however, will remain intact, and so the timeindependent Schrödinger wave-equation that governs quantum entanglement will remain in
place. Because quantum measurements have been carried out on entangled particles in the actual
world there will be slices in wS that are intrinsic duplicates of slices in the actual world at which
wave-packet collapse has occurred. At those slices, there will exist particles with quantum
properties like spin that form systems describable by the wave-equation. For those particles, it
will be possible to project through space to determine information about one particle based on
information about another which, in turn, will support counterfactual dependencies.
Are these counterfactual dependencies causal? We believe that they are. Recall that a
counterfactual counts as causal when it relates distinct events. This case fits the bill. To count as
an event the possession of a quantum property must be predominantly intrinsic and be localised
to a region. The property of having a certain spin is a paradigm intrinsic property. In the actual
world, that property is localised to a spatiotemporal region. So it is an actual event. Moreover,
the possession of a spin property by P1/P2 retains its status of eventhood once we move to wS.
For there is a (spatial) region in wS that is a counterpart of the (spatiotemporal) region occupied
by P1/P2 in the actual world; the relevant slice in wS is itself a counterpart of some hyperplane of
simultaneity in the actual world which is occupied by P1/P2. Hence, the possession of a spin
property counts as an event in both worlds.
To count as causal counterfactuals those events must be distinct. They are. P1’s being spin up
is not a part of P2’s being spin down. Both events are part of the same entangled system, but they
are not parts of each other. Nor does P1’s being spin up logically entail P2’s being spin down. So
P1 and P2 are distinct events in wS, and so the counterfactuals that connect them in that world are
causal.
So much the worse for Lewis’ strategy of cleaving apart the causal and non-causal
counterfactuals, you might say; that strategy needs to be revised. We are doubtful, however, that
15
there is a way to modify Lewis’ account to rule out the entanglement case without also ruling out
a range of standard causal counterfactuals. That’s because the only difference between an
entanglement case and other cases of causal counterfactual dependence between events is that
the former counterfactuals hold even when we strip away time. So the only way to modify Lewis’
strategy for pulling apart causal from non-causal counterfactuals would be to explicitly write
temporality into the conditions under which a counterfactual is causal. That modification,
however, begs the question.
One might, then, abandon the Lewisian project of differentiating causal from non-causal
counterfactuals via the metaphysics of events and turn to the interventionist strategy for
achieving the relevant demarcation advocated by Woodward (2003). But this won’t help: the
interventionist accepts (at least) the distinctness criterion, adding to this the claim that the causal
counterfactuals are the ones involving interventions. The relevant sufficiency condition is stated
as follows:
If (i) there is a possible intervention that changes the value of X such that (ii) carrying out this
intervention (and no other interventions) will change the value of Y ..., then X causes Y.
It follows that if a counterfactual of the form ‘If an intervention set the value of X to x*, then
the value of Y would be y*’ is true, where x* and y* are non-actual values, then X causes Y
(Frisch (2010, p. 665) makes this point). It is relatively straightforward to see that, in wS,
counterfactuals like ~|P1↑> □→ ~|P2↓> and ~|P1↓> □→ ~|P2↑> are causal since they can be
translated into interventionist terms. Suppose that |P1↑>. If an intervention were to be carried
out on P1 such that |P1↓>, then it would follow that |P2↑>. In short: if we intervene to change
P1’s spin property, then this will result in a change to the value of P2’s spin property. Moreover,
since interventions need not be physically possible there is no implication that they are processes
in time. All we require is the logical possibility of carrying out a ‘surgical strike’ on a system, by
16
reaching in and resetting a value whilst holding the rest of the system fixed. Since interventions
in this sense can be carried out on entangled particles, a case can be made for treating
counterfactual dependence of this kind as causal.
4. Objections
We anticipate four objections.
4.1
Semantics
One might deny that quantum entanglement unambiguously gives rise to counterfactual
dependence. Following Glynn and Kroedel (forthcoming) one might argue that whether or not
counterfactual conditionals involving quantum particles are true depends on how we evaluate
those counterfactuals within a relativistic setting. Here’s an example. Suppose, as before, that P1
and P2 are entangled and that P1’s spin is up. Now, consider the counterfactual: ~|P1↑> □→
~|P2↓>. In order to evaluate that counterfactual we need to know which world(s) are closest. We
combine Lewis’ similarity metric with his contention that actually, the future asymmetrically
depends on the past. From this it follows that if we want to counterfactually vary some event, E,
then worlds with pasts qualitatively identical to the actual world up until the counterfactual
variation of E, and different futures from the actual world thereafter, will be closer than worlds
with qualitatively identical futures to the actual world after the counterfactual variation of E, but
different pasts from the actual world. But in a relativistic setting temporal ordering is relative to
an inertial frame of reference. As a consequence there are some frames of reference in a world,
w, such that if the past is qualitatively identical to that of the actual world up until t that entails
holding fixed P2’s spin state, since P2 is in the past relative to P1; mutatis mutandis for P2. This is
not the case for all reference frames: for the frame in which P1 and P2 are simultaneous, it will be
17
possible to apply the standard Lewis-Stalnaker semantics, in the standard way. Still, as Glynn and
Kroedel argue, there is no frame-invariant sense in which P1 depends counterfactually on P2 or
vice versa and so no unambiguous way to say that there is (or is not) counterfactual dependence
between entangled particles.
The alternative is to appeal to another analysis Lewis offers which he takes to be equivalent to
the one we have been deploying. According to that analysis, a counterfactual such as ‘If it had
been the case that P, it would have been the case that Q’, is true iff for some set S of worlds, Q is
true in every world in S. What determines which worlds are in S if not closeness? Suppose that P
encompasses events that occur during the interval Tn. Then every world that is a member of S is
such that:
(i) P is true at w;
(ii) w is exactly like our actual world at all times before Tn;
(iii) w conforms to the actual laws of nature at all times after Tn; and
(iv) during Tn, w differs no more from our actual world than it must to permit P to hold.
In the case under discussion we hold fixed history up to a time t, and then counterfactually
vary P1’s quantum state after t and see what happens to P2’s quantum state. So far, the move
away from Lewis’ closeness metric gets us nothing, since the problem remains: in a
relativistic setting, temporal ordering is relative to an inertial frame of reference. But we can
make one further small amendment to Lewis’ alternative analysis. Rather than holding fixed a
foliated history leading up to P1 (and thus holding fixed a series of slices, one of which may
include P2, depending on the frame of reference), we hold fixed all of the space-time points that
lie on P1’s backwards light cone. We then look to worlds that are similar to the actual world with
regard to the laws of nature, but in which P1’s spin is different and see what changes there are for
18
P2’s spin. As Glynn and Kroedel argue, if we proceed in this fashion then P1 and P2 are
counterfactually dependent upon one another, and thus the result argued for in §3 is retrieved.
Still, Glynn and Kroedel outline a number of modifications of the Lewis-Stalnaker semantics
according to which P1 and P2 are not counterfactually dependent upon one another. Doesn’t
adopting an account of counterfactual dependence according to which entangled particles are
counterfactually dependent upon one another seem unduly self-serving?
We think that of the options that Glynn and Kroedel outline the most natural is one where
we hold fixed P1’s absolute past. After all, what are we trying to do when evaluating such
counterfactuals is hold fixed the causal history of whatever it is we are interested in, such as a
particle or a match-striking. It is for this reason that Lewis advises us to hold fixed the temporal
history of a thing, because in a pre-relativistic setting a thing’s temporal history is likely to be its
causal history. In a relativistic setting, however, most (though not all, if there is non-local
causation) of a thing’s causal history is contained within its absolute past. Or, at least, the closest
analogue to a causal history in a non-relativistic setting is the absolute past in a relativistic setting,
and so holding fixed the absolute past appears to be the most reasonable extension of the LewisStalnaker semantics into relativity.
4.2
Time-Independence
Here’s the second objection. One might deny that wS is a world in which the relevant
quantum mechanical laws hold. Perhaps the laws that govern quantum theory tie time and space
together much more closely than we have assumed. If the time-evolving laws of quantum theory
are necessary for the space-evolving laws then wS would not be a world in which information
about one entangled particle can be used to determine information about its twin.
This objection fails to find its mark. The time-independent Schrödinger wave-equation used
to build the example above is just that: time-independent; it can be used to solve quantum
19
problems in a purely one-dimensional, spatial setting, without time-evolution (for recent
examples, see Vidal (2004), Osborne and Nielsen (2002) and Calabrese and Lefevre (2008)).
Indeed, for the most part the interesting features of quantum mechanics involve its application
to purely spatial cases via the wave equation (Taylor, et al. (2004, pp. 237–238) make this point,
see e.g. Simon and Pan (2002), Navez, et al. (2001) and Sheng and Deng (2010) for some recent
applications). So there is every reason to think that, at the very least, the time-independent
Schrödinger wave-equation will survive the shift from wB to wS in this case, bringing
counterfactual dependence between spatially-distant entangled particles with it.
One might press the point, however, in a slightly different way. While it is true that the timeindependent Schrödinger wave-equation is time-free, perhaps the collapse of the wave-packet is
not a time-free phenomenon. There are two things to say in response. First, it is not obvious that
the collapse of the wave packet requires time. We know through experimentation that the speed
of the wave packet collapse is, at least, 10,000 times the speed of light. This is unsurprising since,
mathematically, the collapse is treated as instantaneous. Second, a number of the dominant
interpretations of quantum mechanics give an account of the wave packet collapse that does not
support the idea that it requires time. On the Copenhagen interpretation, the collapse of the
wave-packet is not a physical process. It is, rather, a merely epistemic phenomenon, one that
reflects our ignorance about the quantum states that entangled particles like P1 and P2 possess.
Because the collapse of the wave-packet is not a physical process it makes no sense to think of
time emerging out of such a collapse. Things are even worse on the many worlds interpretation,
according to which there just is no collapse of the wave packet.
The issue of what to make of the wave packet collapse goes to the heart of some vexing
concerns in quantum theory. But that works in our favour: because of how controversial wave
packet collapse is, addressing our argument by taking a stand on this issue is a risky strategy.
4.3
Common Cause
20
According to the third objection the entangled particles, P1 and P2, while counterfactually
dependent upon one another, are not causally related. Instead there is a common cause in the
particles’ past that caused them to be entangled and thus counterfactually dependent upon one
another.
Now generally, that a common cause exists for some x and y does not obviate the possibility
that x is causally dependent on y. Suppose a monk rings a gong at t1. This causes a second monk
to ring a second gong at t2. At t3 a third monk will ring her gong only when she hears two gongs
sounded, one after another. Here the second gong is a cause of the third monk’s ringing the third
gong and the first gong is a common cause of both the second gong ringing and the third. The
entanglement case is similar: both P1 and P2 owe their entanglement to some prior event and they
are involved in a causal counterfactual relationship with each other. The only difference is that
the entanglement event is a common cause, but not also a joint cause of either P1 or P2’s spin
property since P1’s having a spin property is sufficient for P2’s spin property (so long as we have
the time-independent Schrödinger equation in place).
One might press the point by arguing that the importance of entanglement as a common
cause poses a problem for our claim that, given a counterfactual theory of causation, there can
be causation without time. On our picture P1 and P2 become entangled at t1 and the wave packet
collapse occurs at t2. The entanglement, at t1, could be thought of as a common cause (across
time) of P1 and P2’s spin properties. In a world (wS) in which the B-relations have been stripped
out, you are likely to lose the relevant instance of across time causation. Then, one might argue it
is not true that P1 and P2 are entangled.
This is a version of the previous objection: that time is needed for the counterfactual
dependence of P1 on P2 and vice versa. As such, it can be handled in a similar fashion. The world
we are imagining is an intrinsic slice duplicate. It is therefore a world in which all of the intrinsic
facts about P1 and P2 are preserved at the slice that corresponds to t2. Thus, if P1 and P2 have
21
determinate spin properties at t2, then these will be duplicated when we move to the new world.
Similarly, everything that goes on, intrinsically, with entanglement will be duplicated. It is just
that the slices corresponding to t1 and t2 won’t be temporally related. In such a world we might
lose causation between these slices, but we will retain all of the relevant intrinsic facts plus the
laws needed for the causal counterfactuals between P1 and P2. That’s because, again, spin
properties are intrinsic and the time-independent Schrödinger equation is just that: timeindependent.
4.4
Locality
Fourth objection: causation is local! There is no such thing as non-local causation, or ‘spooky
action at a distance’. This is a problem because wS is a world in which there is causation only if it
is a world in which there is non-local causation. That the counterfactual theory of causation
treats entanglement as causal might be turned into a reductio of the counterfactual theory because
of its commitment to non-locality. But that would be too quick. A central part of the worry
about non-local causation concerns the notion of physical influence. Suppose the causation of x
by y implies the existence of some relation of physical influence between x and y, perhaps some
step-wise physical process linking the two events. Then if the counterfactual theory implies the
existence of causation between entangled particles it also implies the existence of some physical
process linking the particles. But such a process would need to be superluminal and as yet we
have no substantial empirical evidence that anything travels faster than the speed of light.
But if that’s why the implication from the counterfactual theory of causation to the existence
of non-local causation is worrying then we should not be swayed. One benefit of a
counterfactual theory of causation is that it can get by without the need for underlying physical
processes of this kind. This is most evident in cases of absence causation which involves the
causation of some event y by the lack of some event x. It is implausible to suppose that in such
22
cases there is always a physical process underpinning the relevant instance of causation.
Nevertheless, many find it is natural to suppose that there is causation in such cases. The
counterfactual theory is capable of handling these intuitions precisely because actual physical
processes are not required for it to be true that x causes y. So we should not be tempted by the
thought that the non-local causation implied by the counterfactual theory thereby implies
superluminal processes.
5.
Conclusion
We have argued that, given a counterfactual theory of causation, the relationship between
entangled particles is causal. This result can be parlayed into an argument for the view that time
is not necessary for causation. The upshot is that, if the counterfactual theory of causation is
true, there can be causation without time. As noted from the outset we foresee two morals one
might draw: embrace the result or use it as a counterexample to the counterfactual theory. We
would like to close by emphasising the two advantages associated with embracing the result.
First, it opens up a way to recover causation in the face of physical and metaphysical theories
that eliminate time, which is a central step along the road to recovering the appearance as-of
temporality in a timeless world. Second, it presents a way to understand the relationship between
entangled particles. That relationship is genuinely causal, but not ‘spooky’ because no non-local
physical influences are required.
We recognise that in both cases these might be seen as disadvantages. If one thinks that
causation is necessary for time or that non-local causation is out of the question, then the
implications that we have identified for a counterfactual approach constitute new reasons to
doubt that view. Clearly, that is not where our sympathies lie, but what really matters for present
purposes is that we have laid bare a bit more of the conceptual terrain underlying the
23
counterfactual theory of causation, with the aim of sparking further discussion over the
relationship between causation and time.
Works Cited
Barbour, J. (1999). The End of Time. (Oxford; New York: Oxford University Press).
Calabrese, P., and Lefevre, A. (2008) Entanglement Spectra in One-Dimensional Systems.
Physical Review A, 78(3), 032329.
Coady, D. (2004). Preempting Preemption. In Causation and Counterfactuals. John Collins, Ned
Hall and Laurie A. Paul (Eds.), 325–340. (Cambridge, M.A.: MIT Press).
Collins, J., Hall, N., and Paul, L. A., (Eds.) Causation and Counterfactuals. (Cambridge, M.A. : MIT
Press, 2004).
Dowe, P. (1992) Wesley Salmon's Process Theory of Causality and the Conserved Quantity
Theory. Philosophy of Science, 59(2), 195–216.
Frisch, M. (2010) Causes, Counterfactuals, and Non-Locality. Australasian Journal of Philosophy,
88(4), 655–672.
Glynn, L. & Kroedel, T. (forthcoming) Relativity, Quantum Entanglement, Counterfactuals and
Causation. British Journal for the Philosophy of Science.
Hall, N. (2004). Two Concepts of Causation. In Causation and Counterfactuals. John Collins, Ned
Hall and Laurie. A. Paul (Eds.), 225–276. (Cambridge, M.A.: MIT PRess).
Healey, R. (2002). Can Physics Coherently Deny the Reality of Time? In Time, Reality &
Experience. Craig Callender (Ed.), 293–316. (Cambridge: Cambridge University Press).
Lewis, D. (1973a) Causation. Journal of Philosophy, 70(17), 556–567.
Lewis, D. (1973b). Counterfactuals. (Oxford: Basil Blackwell).
Lewis, D. (1979) Counterfactual Dependence and Time's Arrow. Noûs, 13(4), 455–476.
Lewis, D. (1986) Events. In Philosophical Papers Vol. 2, 241–269. (New York: Oxford University
Press).
Menzies, P. (1996) Probabilistic Causation and the Pre-Emption Problem. Mind, 105(416), 85–
117.
Navez, P., Brambilla, E., Gatti, A., and Lugiato, L. A. (2001) Spatial Entanglement of Twin
Quantum Images. Physical Review A, 65(1), 013813.
Osborne, T. J., and Nielsen, M. A. (2002) Entanglement in a Simple Quantum Phase Transition.
Physical Review A, 66(3), 032110.
24
Pusey, M. F., Barrett, J. & Rudolph, T. (2012) On the Reality of the Quantum State, Nature
Physics 8(6), 475–478.
Rovelli, C. (2007). The Disappearance of Space and Time. In The Ontology of Spacetime. Dennis
Dieks (Ed.), 25–36. (Amsterdam: Elsevier).
Salmon, W. (1994) Causality without Counterfactuals. Philosophy of Science, 61(2), 297–312.
Schaffer, J. (2000) Causation by Disconnection. Philosophy of Science, 67(2), 285–300.
Schaffer, J. (2005) Contrastive Causation. Philosophical Review, 114(3), 327–358.
Schlosshauer, M., and Fine, A. (2012) Implications of the Pusey-Barrett-Rudolph Quantum NoGo Theorem. Physics Review Letters 108(26), 260404.
Sheng, Y.-B., and Deng, F.-G. (2010) One-Step Deterministic Polarization Entanglement
Purification Using Spatial Entanglement. Physical Review A, 82(4), 044305.
Simon, C., and Pan, J.-W. (2002) Polarization Entanglement Purification Using Spatial
Entanglement. Physics Review Letters, 89(25), 257901.
Skow, B. (2007) What Makes Time Different from Space? Noûs, 41(2), 227–252.
Tallant, J. (2008) What Is It to "B" a Relation? Synthese, 162, 117–132.
Taylor, J. R., Zafirators, C. D., and Dumbson, M. A. (2004). Modern Physics for Scientists and
Engineers. Pearson Prentice Hall).
Vidal, G. (2004) Efficient Simulation of One-Dimensional Quantum Many-Body Systems.
Physical Review Letters, 93(4), 040502.
Woodward, J. (2003). Making Things Happen: A Theory of Causal Explanation. (Oxford: Oxford
University Press).
25