## 1. Regularity Theories of Causation



The core idea of regularity theories of causation is that causes are
regularly followed by their effects. A genuine cause and its effect
stand in a pattern of invariable succession: whenever the cause
occurs, so does its effect. This regular association is to be
understood by contrast to a relation of causal power or efficacy. On a
regularity theory, a cause and its effect only instantiate a
regularity. No causal powers are posited in virtue of which a cause
brings about its effect. Moreover, no metaphysical entities or
connections are posited which would ground the regularities in the
world. Causation is not conceived of as a metaphysically thick
relation of production, but rather as simple instantiation of
regularity.


### 1.1 Humean Regularity Theory



The most influential regularity theory is due to Hume. He defines a
cause to be



> 
> 
> [a]n object precedent and contiguous to another, and where all the
> objects resembling the former are plac'd in like relations of
> priority and contiguity to those objects, that resemble the latter.
> (Hume 1739: book I, part III, section XIV [1978: 169])
> 
> 
> 



This definition contains three conditions for a cause. First, a cause
temporally precedes its effect. Second, a cause is contiguous to its
effect. That is, a cause is in spatiotemporal vicinity to its effect.
Third, all objects similar to the cause are in a "like
relation" to objects similar to the effect. This third
resemblance condition says that cause and effect instantiate a
regularity. To see this, note the presupposition of the resemblance
condition: causes and effects can be sorted into types. The objects
which are like the cause are of a certain type and so are the objects
which are like the effect. The third condition thus says that all
objects of a certain type are followed by an effect of a certain
type.



A Humean Regularity Theory (HRT), which may or may not coincide with
Hume's genuine analysis of causation, can be summarized as
follows: where \(c\) and \(e\) denote token events of the types \(C\)
and \(E\), respectively,



\(c\) is a cause of \(e\) iff



(i)
\(c\) is spatiotemporally contiguous to \(e\),
(ii)
\(c\) precedes \(e\) in time, and
(iii)
all events of type \(C\) are followed by an event of type
\(E\).


Causation is thus analysed in terms of spatiotemporal contiguity,
temporal precedence, and regular connection. The causal relation is
reduced to non-causal entities: the two particular facts about
spatiotemporal contiguity and temporal precedence, and the general
regularity. There is nothing more to causation according to HRT. In
particular, causation does not involve a necessary connection, a
productive relation, causal powers, or the like--not even to
ground the regularities. This stance against metaphysically thick
conceptions of causation is characteristic for the regularity theory
(see, e.g., Dowe 2000: Ch. 2; Psillos 2009).



Let us explain some of the corollaries of HRT. The spatiotemporal
contiguity of cause and effect excludes causation at a distance: a
cause is proximate to its effect, either directly or via a chain of
contiguous events (Hume 1739: book 1, part III, section II [1978:
75]). The temporal precedence of the cause and the direction of time
imply that causation is asymmetric: if \(c\) causes \(e\), then it is
not the case that \(e\) causes \(c\). Temporal precedence thus
explains the asymmetry of causation, and how we can distinguish
between cause and effect. However, it also rules out the possibility
of simultaneous and backwards-in-time causation; and it dampens the
prospects of a non-circular theory of time defined in terms of
causation.



Condition (iii) relates a particular instance of causation, \(c\) is a
cause of \(e\), to the class of all like cases, the types of \(c\) and
\(e\). Hume takes causation to be primarily a relation between
particular matters of fact. Yet the causal relation between these
actual particulars holds in virtue of a certain regularity. Any causal
relation thus supervenes on a regularity in the sense that there can
be no change in the causal relation without a change in the
regularity. Any regularity, in turn, supervenes on particular matters
of fact: there can be no change in the regularity without a change of
particular matters of fact. Whether a regularity is true is thus
determined by which particular matters of fact obtain. Likewise,
whether a causal relation obtains is determined by which regularity
obtains. In the end, only the particular matters of fact determine
whether or not a causal relation obtains, since the relation of
supervenience is transitive. (For details, see the entries on
 supervenience
 and
 David Lewis, section on Humean supervenience.)



According to condition (iii), a cause is a *sufficient*
condition for its effect in this sense: *all* \(c\)-like events
occurring in the actual world are in fact followed by an \(e\), where
the actual world is understood as the collection of all particular
spatiotemporal matters of facts. The sufficiency of the cause for the
effect is not to be understood in a modally stronger sense (Dowe 2000:
20). In particular, Hume denies that a cause necessitates its effect.
In the world, there is--to use Hume's phrase--just a
"constant conjunction" between cause and effect.



On HRT, every effect has a sufficient cause. This does not imply that
any event has a sufficient cause, but it says that, if there is a
causal relation, it is deterministic. Probabilistic causation is thus
not covered by HRT. And while the theory is compatible with a
probabilistic analog of causation, we follow Hume in restricting
ourselves to deterministic causation in this entry. (See the entry on
 probabilistic causation
 for an overview of theories of causation, where the underlying
processes may be indeterministic.)



HRT allows us to discover causal relations. We may observe
spatiotemporal contiguity, temporal precedence, and that a certain
type of event is generally followed by another type of event. Upon
repeated observation of an invariable pattern of succession, we begin
to expect that this particular event of a certain type is followed by
that particular event of a certain type. We develop an inferential
habit due to the experienced constant conjunction of types. Or as Hume
puts it:



> 
> 
> A *cause* is an object precedent and contiguous to another, and
> so united with it, that the idea of the one determines the mind to
> form the idea of the other. (Hume 1739: book 1, part III, section XIV
> [1978: 170])
> 
> 
> 



Our mind forms a habit to expect or to infer the effect from a cause
upon repeatedly experiencing events which occur in sequence. Thereby,
we produce a connection between objects or events that goes
"beyond what is immediately present to the senses" (1739:
book I, part III, section II [1978: 73]). Our senses and our memory
provide us only with finitely many instances of objects or events
which occur jointly, and from which our minds form inferential habits.
Still these inferential habits apply also to future, as of yet
unobserved instances of the regularity. In Hume's words,



> 
> 
> after the discovery of the constant conjunction of any objects, we
> always draw an inference from one object to another, [...].
> Perhaps 'twill appear in the end, that the necessary connexion
> depends on the inference, instead of the inference's depending
> on the necessary connexion. (Hume 1739: book 1, part III, section VI
> [1978: 88])
> 
> 
> 



And indeed, Hume says that "[n]ecessity is something that exists
in the minds, not in objects" (1739: book I, part III, section
XIV [1978: 165]). So there is no necessary connection *in the
world* which would go beyond regular association. However, the
inferential habits we develop based on the perceived regularities
makes us feel that the connection between cause and effect is stronger
than mere invariable succession. The necessary connection exists
*in our minds* but *not in the objects* (see section in
the entry on
 Hume on Necessary Connection: Constructive Phase).



According to the standard interpretation, the regularities themselves
are real. There really are jointly and repeatedly occurring events of
certain types that are spatially contiguous and some temporally
precede others. And the regularities are mind-independent: even if
there were no minds around, the regularities would exist as patterns
in the Humean mosaic of particular facts.



Is causation in the world, as expressed by HRT, extensionally
equivalent to causation in the mind? The difference between the two
definitions seems to be that condition (iii) of HRT is replaced by the
following condition: \(c\) of type \(C\) makes us infer \(e\) of type
\(E\). While HRT does not require the presence of an epistemic agent,
the other definition seems to require such an agent and is thus
mind-dependent. And an epistemic agent may infer an event from another
event, even though there is no regularity connecting the two events.
Having said that the two definitions are not extensionally equivalent,
the definitions can be read in ways which make them coextensive. The
epistemic agent of the second definition, for example, can be seen as
a hypothetical omniscient observer who always and only infers in
accordance with all and only true regularities (for details, see
Garrett 1997: Ch. 5).



What is Hume's view on causation? We will not enter this
exegetical debate here. Suffice it to say that the interpretations of
Hume are manifold. Psillos (2009: 132) says that HRT "was
Hume's view, too, give or take a bit" and argues for it in
Psillos (2002: Ch. 1). But there are many deviating interpretations
(Beebee 2006; Garrett 2009; Strawson 1989; J. Wright 1983). The
relation between causation in the world and causation in the mind, in
particular, has remained a matter of debate among Hume scholars (see
e.g., Garrett 1993).



Let's go back to the Humean Regularity Theory. HRT enjoys two
benefits. As we have observed, no substantive metaphysics is needed to
explain causation. No entities like causal powers or a necessary
connection between cause and effect need to be posited. Causation is
rather reduced to spatiotemporal contiguity, temporal precedence, and
regular association. Second, Hume's regularity theory explains
how we can figure out what causes what. A cause is simply the event
which temporally precedes and is spatiotemporally contiguous to
another event such that the former and latter event instantiate a
regularity. Since temporal precedence and spatiotemporal precedence
can be observed, the problem of identifying causes reduces to the
problem of identifying regularities.



However, HRT also faces serious problems. Recall that it presupposes
that causes and effects can be sorted into types which are connected
by a regularity. But it is unclear how the sorting can be achieved.
What makes an event "like" another? Which aspects of the
objects or events are relevant for the sorting? Many contemporary
philosophers explicate the relevant notion of resemblance or
similarity in terms of laws of nature. If there is a law of the form
\(\forall x (A(x) \rightarrow B(x))\), then events similar to a token
event \(a\) are simply events of the same type--other
instantiations of \(A\)--and events similar to \(b\) are also
just token events of the same type. However, it is a non-trivial task
to give a satisfying account of what a law of nature is--to say
the least.



A second problem runs as follows. Assume, plausibly, that a giant
meteor caused the dinosaurs to go extinct. For the meteor to be a
cause on HRT, there must be a regularity saying that events like giant
meteors hitting the Earth are followed by events like the extinction
of the dinosaurs. But it seems as if such a regularity does not exist:
the next giant meteor hitting the Earth will not cause the dinosaurs
to go extinct. The dinosaurs are already extinct. The problem is that
the meteor is only a cause of the extinction on a single occasion. But
according to HRT this one meteor hitting the Earth is only a cause if
some other meteors hitting the Earth also satisfy the regularity. But
why would events far away in space and time determine whether or not
this giant meteor caused the extinction of the dinosaurs on this
occasion? It seems as if we can have causation without regularity on a
single occasion. We will refer to this problem as the problem of
singular causal relations.



A third problem is that there can be regularity without causation. The
cry of the cock precedes the rise of the sun and the two events are
regularly associated. And yet the cry of the cock is not a cause of
the sunrise. A special case of this problem arises from joint effects
of a common cause. Suppose an instantiation \(c\) of \(C\) is a common
cause of the instantiations \(a\) of \(A\) and \(b\) of \(B\), where
the token event \(a\) precedes the token event \(b\) in time (see
 Figure 1).
 Further suppose \(C\) is instantiated whenever \(A\) is. That is,
whenever a token event of type \(A\) occurs, a token event of type
\(C\) occurs. Then there is a regular connection between \(A\) and
\(B\), which we normally do not count as a causal relation. This
problem remains a central challenge for regularity theories to
date.



![a diagram with three disjoint circles labeled A, B, and C respectively. An arrow points from circle C to circle A and another from circle C to circle B](figure1.svg)

Figure 1




Setting aside the possibilities of remote and backwards causation for
now, a regularity theory of causation is only tenable when the three
problems can be answered. The next section focuses on how laws of
nature may help overcome the problem of sorting events into
appropriate types and how they may exclude the cry of the cock as a
cause of the sunrise.


### 1.2 Regularities and Laws



Mill (1843) refined the Humean Regularity Theory of causation. An
effect usually occurs only due to several instantiated factors, that
is, instantiated event types. He distinguishes between positive and
negative factors. Let \(C\_1,\) \(C\_2,\) ..., \(C\_n\) be a set of
positive factors and \(\overline{D}\_1,\) \(\overline{D}\_2,\) ...,
\(\overline{D}\_m\) a set of negative factors which are together
sufficient for an effect \(E\). Mill says that the totality of present
positive and absent negative factors sufficient for an effect to occur
is the cause of this effect. In symbols, the totality of the token
events \(c\_1,\) \(c\_2,\) ..., \(c\_n\) and the absence of any
token events of the types \(D\_1,\) \(D\_2,\) ..., \(D\_m\) is a
cause of an occurring event \(e\) iff \(C\_1,\) \(C\_2,\) ...,
\(C\_n,\) \(\overline{D}\_1,\) \(\overline{D}\_2,\) ...,
\(\overline{D}\_m\) is sufficient for \(E\).



Furthermore, Mill claims that causation requires lawlike regularities.
Mere regular association is not enough. The cry of a cock (as part of
a totality of conditions) precedes the rise of the sun and the former
is invariably followed by the latter, and yet the cry of a cock does
not count as a cause of the sunrise. For the regularity is not a law
of nature, it is a mere accidental regularity.



The Millian Regularity Theory (MRT) can be summarized as follows:



The totality of present positive factors \(\{ C\_{1-n} \}\) and absent
negative factors \(\{ \overline{D}\_{1-m} \}\) is a cause of an event
\(e\) of type \(E\) iff



(i)


there are events of type \(C\_1,\) \(C\_2,\) ..., \(C\_n\) which are
spatiotemporally contiguous to \(e\) and there are no such events of
type \(D\_1,\) \(D\_2,\) ..., \(D\_m\),

 
(ii)


the events of type \(C\_1,\) \(C\_2,\) ..., \(C\_n\) precede \(e\)
in time, and

 
(iii)


it is a *law of nature* that an instantiation of all positive
factors \(\{ C\_{1-n} \}\) is followed by an instantiation of factor
\(E\) when none of the negative factors \(\{ D\_{1-m} \}\) is
instantiated.

 


What are laws of nature? Well, on a regularity theory of causation, a
law is a regularity understood as a stable pattern of
events--nothing more, nothing less. And yet, laws are still
special regularities. According to Mill (1843), regularities which
subsume other true regularities are more general than the subsumed
ones. The laws of nature are those regularities which are the most
general. They thus organise our knowledge of the world in an ideal
deductive system which strikes the best balance between simplicity and
strength (see, e.g., 1843: Book III, Ch. XII). This best system
account of laws has been further developed by Ramsey and Lewis. Ramsey
puts it as follows,



> 
> 
> even if we knew everything, we should still want to systematize our
> knowledge as a deductive system, and the general axioms in that system
> would be the fundamental laws of nature. The choice of axioms is bound
> to some extent to be arbitrary, but what is less likely to be
> arbitrary if any simplicity is to be preserved is a body of
> fundamental generalizations, some to be taken as axioms and others
> deduced. (1928 [1978: 131])
> 
> 
> 



Everything else being equal, a system with fewer axioms is simpler.
Simplicity thus demands the pruning of inessential elements from the
system of laws. A deductive system is stronger when it entails more
truths. Strength demands that the deductive system of laws should be
as informative as possible. Note that simplicity and strength compete.
A system consisting of only one uncomplicated axiom is simple but not
very informative. A system which has one axiom for any truth is very
strong but far from simple. To avoid a deductive system that is too
simple to be informative and one that is too complicated, we need to
balance simplicity and strength. The proper balance between simplicity
and strength makes the deductive system best.



There is no guarantee that there is a unique best deductive system. If
there is none, the laws of nature are those axioms and theorems that
are common to all deductive systems that tie with respect to
simplicity and strength (or are at least good enough). D. Lewis thus
says that



> 
> 
> a contingent generalization is a *law of nature* if and only if
> it appears as a theorem (or axiom) in each of the true deductive
> systems that achieves a best combination of simplicity and strength.
> (1973: 73)
> 
> 
> 



As a consequence, laws are simple and yet highly informative as a part
of each best deductive system. The regularity about the cry of the
cock and the sunrise is not part of a best deductive system. And
whichever regularity is not part of any best system is accidental: it
is not a law of nature. The dividing line between laws and accidental
regularities is determined by being a part of any best system. Note
that no regularity can be regarded as a law of nature in absence of a
full system.



There is a worry that the best system account of laws is infected by
epistemic considerations. Of course, what truths a best system entails
is an objective matter, it does not depend on our knowledge of it.
Whether a truth is objectively implied by a best system, however,
depends on the way this system is organized. And there may be
considerable leeway in choosing the axioms to obtain a simple system.
The worry is that the notion of simplicity is not fully objective, and
even if it were, striking a balance between simplicity and strength is
not. Hence, which regularities are laws depends on epistemic criteria,
our desiderata to organize our knowledge as simple as possible in a
deductive system. This is not to say that the lawlike regularities are
mind-dependent. Laws and regularities are still patterns in the world
independent of our knowledge of them. What makes some of the
regularities laws and some others not, however, is not only determined
by the world. The law-making feature, striking a proper balance
between simplicity and strength, seems to have an epistemic component.
This is presumably the cost for being free of substantive metaphysical
commitments. (For details on the best system account of laws, see
the entry on
 laws of nature.)



What does the epistemic component mean for causation as lawlike
regularity? The regularity is determined solely by the world, but the
lawlikeness of a regularity rests, in part, on epistemic criteria. So,
while causation is not arbitrary--there must still be a
regularity for causation--there is an epistemic element in
choosing which regularities constitute causation. In brief, the
regularities are mind-independent, what causes what is not.



The Humean regularity theory, where causation is instantiation of
regularity, presupposes that events can be sorted into types. A type
is a class of resembling events. Based on the best systems account, we
may now appeal to laws of nature in order to determine which events
are similar and which are dissimilar. Suppose it is a law that all
events \(a\) of type \(A\) are followed by events \(b\) of type \(B\).
Then the events or objects similar to \(a\) are just the other
instantiations of type \(A\), and likewise for type \(B\). But the
law-making feature depends on epistemic considerations, and so the
types determined by those laws depend on epistemic considerations.
Hence, falling back to a Humean regularity theory does not solve the
issue. We still need a way to sort events into types. But which events
resemble each other comes in degrees and depends on the respects of
comparison. The degrees of similarity and respects of comparison are
not fixed by the world understood as the collection of all particular
spatiotemporal matters of fact. The degrees and respects are rather
epistemic components which depend on the categories and classification
schemes we use. A related problem are gerrymandered categories or
properties like "grue", as studied by Goodman (1955) in
his new riddle of induction.



Some defenders of the regularity theory see the dependence on
epistemic components as a problem. To solve it, one may want to posit
that there are natural properties which carve nature at its joints.
The idea is that those natural properties provide a fully objective
classification. Those natural properties would ensure that the lawlike
regularities are free of epistemic components. However, what is
natural is notoriously difficult to define. Natural properties, which
are objectively similar and dissimilar to each other, are thus often
simply posited as primitives which are not analysed further, as Lewis
(1986: 60-2) does for example.


### 1.3 INUS Conditions



On the Millian Regularity Theory, a cause is nomologically sufficient
for its effect: there is a law of nature such that the occurrence of a
cause suffices for the occurrence of its effect. A cause is taken to
be a totality of positive and negative factors which is sufficient and
necessary for the effect. However, one and the same effect can have
*many* causes understood as sufficient and necessary totalities
(as has already been observed by Venn 1889). Each of these distinct
totalities, or clusters of factors, is sufficient to bring about the
effect, but no single one is necessary if another cluster is
actual.



Mackie (1974: 63) conceives of each cluster of factors as a
conjunction, for example \(C \land \neg A\), where \(C\) is a type of
event and \(\neg A\) is the absence of any event of type \(A\).
Consider, for instance, a house which burns to the ground. There is a
variety of distinct clusters which can cause the burning of the house.
One cluster includes a short-circuit, the presence of oxygen, the
absence of a sprinkler system, and so on. Another cluster includes an
arsonist, the use of petrol, again the presence of oxygen, and so on.
Yet another cluster includes a burning tree falling on the house, and
so on. Each cluster is a conjunction of single factors. The
disjunction of all clusters represents then the plurality of causes
(see Mackie 1965: 245 and 1974: 37-38 for details).



The disjunction of all clusters for an effect is relative to a
*causal field*. A causal field is "a background against
which the causing goes on" (1974: 63). Roughly, a causal field
captures the circumstances which are kept fixed and so cannot even be
considered as (part of) a cause. Or, as Mackie puts it, what is caused
is



> 
> 
> not just an event, but an event-in-a-certain-field, and some
> conditions can be set aside as not causing this-event-in-this-field
> simply because they are part of the chosen field. (1974: 35)
> 
> 
> 



Suppose a causal field includes that a certain person was born and
alive for a while. Being born thus cannot be a cause of the
person's death relative to this causal field. However, with
enough ingenuity, the person being born is a cause of the
person's death relative to a causal field which does not include
that the person was born and alive for a while as a part of the fixed
background (see 1974: 37).



This suggests that the regularity for an effect is a disjunction of
conjunctions which is necessary and sufficient for the effect relative
to a causal field. Relative to this field, each cluster is sufficient
for the burning of the house, but no single one is necessary; for
another cluster can be sufficient for the burning. (\(C\) is
sufficient for \(E\) means that, whenever an event of type \(C\)
occurs, so does an event of type \(E\). \(C\) is necessary for \(E\)
means that an event of type \(E\) only occurs if an event of type
\(C\) occurs.) For simplicity, suppose the complex regularity
(relative to a particular causal field) has the form:

 

\[(C\_1 \land C\_2) \lor D\_1 \leftrightarrow E.\]


On the one hand, any cluster, for example \(C\_1 \land C\_2\), is
*minimally sufficient* for \(E\) (and so is the
"cluster" \(D\_1\)). \(C\_1 \land C\_2\) is minimally
sufficient for \(E\) means that the cluster is sufficient for \(E\)
and each conjunct of the cluster is necessary for its sufficiency. So
no conjunct of any cluster is redundant for bringing about the effect.
On the other hand, neither cluster is necessary for the effect \(E\).
\(C\_1 \land C\_2\) is minimally sufficient but not necessary for \(E\).
\(E\) occurs if \(D\_1\) does. Now, focus on \(C\_1\) which is on its
own insufficient to bring about \(E\). \(C\_1\) is a single factor of
the cluster \(C\_1 \land C\_2\) which is a sufficient but unnecessary
condition for \(E\). Hence, \(C\_1\) is an *insufficient but
non-redundant part of an unnecessary but sufficient condition* for
\(E\) (relative to a causal field). Or simply, \(C\_1\) is an INUS
condition for \(E\).



In general, a complex regularity has the form:

 

\[(C\_{1,1} \land \ldots \land C\_{1, n}) \lor \ldots \lor (C\_{k, 1} \land \ldots \land C\_{k, m}) \leftrightarrow E.\]


Each single factor \(C\_{i, j}\) of a cluster is a potential cause.
Hence, a cause is at least an INUS condition (relative to a particular
causal field). "At least" because there are limiting
cases:



(i)
if the regularity has the form \[C\_{1} \lor C\_{2} \leftrightarrow E,\] the cause \(C\_1\) is a sufficient condition for \(E\);
(ii)
if the regularity has the form \[C\_{1} \land C\_{2} \leftrightarrow E,\] the cause \(C\_1\) is a necessary condition for \(E\); and
(iii)
if the regularity has the form \[C\_{1} \leftrightarrow E,\] the cause \(C\_1\) is a necessary and sufficient condition for \(E\).


On Mackie's theory, a factor \(C\) is a cause of \(E\) iff \(C\)
is at least an INUS condition of \(E\), and each factor of the cluster
that contains \(C\) and is sufficient for \(E\) is instantiated.



How does Mackie's theory of causation go beyond the regularity
theories by Hume and Mill? Unlike HRT, Mackie's theory captures
causal scenarios where several different events are necessary to bring
about a certain effect, and so qualify as a cause of the latter. Take
the regularity

 

\[(C\_1 \land C\_2) \lor D\_1 \leftrightarrow E.\]


Suppose \(C\_1, C\_2,\) and \(D\_1\) occur independently of one another.
There is neither a strict positive nor a strict negative correlation
among these events. There are cases where \(C\_1\) does not occur but
\(D\_1\) does, and so \(E\) will occur. There are also cases where
\(C\_1\) occurs but \(C\_2\) and \(D\_1\) do not, and so \(E\) will not
occur. And there will be cases where \(C\_1\) and \(C\_2\) occur
together so that an instance of \(C\_1\) qualifies as a cause of an
instance of \(E\). An event of type \(C\_1\) may therefore be a cause
of an event of type \(E\), without there being a strict regular
succession between \(C\_1\)-events and \(E\)-events. This being said,
Hume (1739: book I, part III, section XV) considers causation without
strict regular succession. There, he seems to propose a regularity
theory that goes beyond HRT and may cover causation based on
non-strict succession.



The advantages of Mackie's theory over Mill's are less
obvious. Recall that for Mill, a cause is a totality of positive and
negative factors such that, in the respective causal scenario, this
totality is sufficient for \(E\). On this theory, events of type \(C\)
may be members of a cause for an effect of type \(E\), without there
being a strict regular succession between \(C\)-events and
\(E\)-events. Moreover, the notion of a totality of factors is
consistent with there being several totalities that are sufficient to
bring about an event of type \(E\). In other words, several totalities
may serve as instances of some cluster in one and the same
Mackie-style regularity. These totalities may be represented by
conjunctions and joined by disjunctions. If we amend the Millian
regularity theory by a requirement of minimal sufficiency, it seems as
if we can go back and forth between a Mill-style and a Mackie-style
representation of causes, with the qualification that causes in the
sense of Mackie are mere factors of a cause in the sense of Mill. In
this sense, the two theories seem to be intertranslatable. It should
be pointed out, however, that Mackie's theory gives us a more
explicit and concise representation of the several totalities, or
clusters, which are minimally sufficient to bring about a certain type
of effect. The complex regularities and their elegant logical
representation have not been in the conceptual repertoire of Mill.



Another merit of Mackie's theory is that it provides an
explanation for causal inference. If we know that an effect of type
\(E\) occurred, and that \(D\_1\) was not present, we may infer that
the cluster \((C\_1 \land C\_2)\) occurred. In particular, we may infer
that \(C\_1\) caused \(E\).



Mackie's regularity theory of causation still fails to
distinguish between genuine causes and mere joint effects of a common
cause. To see this, consider an example adapted from Russell (1921
[1961: 289]): the sounding of factory hooters in Manchester is
regularly followed by London workers leaving their work. And yet the
former is not a cause of the latter. However, as Mackie (1974:
81-84) himself points out, the sounding of the hooters in
Manchester counts as a cause for the workers going home in London:



> 
> 
> In more concrete terms, the sounding of the Manchester factory
> hooters, plus the absence of whatever conditions would make them sound
> when it wasn't five o'clock, plus the presence of whatever
> conditions are, along with its being five o'clock, jointly
> sufficient for the Londoners to stop work a moment
> later--including, say, automatic devices for setting off the
> London hooters at five o'clock, is a conjunction of features
> which is unconditionally followed by the Londoners stopping work.
> (1974: 84)
> 
> 
> 



The structure of the example can be given as follows. There are two
instantiated effects \(E\_1\) (the sounding of the hooters in
Manchester) and \(E\_2\) (the workers going home in London) such that
\(C\_1\) (knocking-off time at five o'clock) is an instantiated
common INUS condition:

 

\[\begin{align} (C\_1 \land C\_2) & \lor D \leftrightarrow E\_1, \text{ and}\\ (C\_1 \land C\_3) & \lor B \leftrightarrow E\_2. \end{align} \]


Furthermore, let's say that \(C\_2\) and \(C\_3\) are instantiated
on this particular occasion. Well then, \(C\_1\) counts as a common
cause of \(E\_1\) and \(E\_2\) on Mackie's theory.



In the presence of the complex regularities, the cluster



\[E\_1 \land \neg D \land C\_3\]


is sufficient for \(E\_2\). Indeed, the cluster of the sounding of the
hooters in Manchester (\(E\_1\)), the absence of any conditions that
would make the Manchester hooters sound when it was not five
o'clock (\(\neg D\)), and the presence of all conditions
(\(C\_3\)) that are together with it being five o'clock (\(C\_1\))
jointly sufficient for the workers going home in London
(\(E\_2\))--this cluster is sufficient for the workers going home
in London. Hence, \(E\_1\) is at least an INUS condition of \(E\_2\). If
\(D\) is not instantiated, but \(C\_2,\) \(C\_3,\) \(E\_1,\) and \(E\_2\)
are, then the instantiation of \(E\_1\) counts as a cause of the one of
\(E\_2\). More generally, an effect of a cause \(C\_1\) may be part of
an instantiated cluster sufficient for another effect of \(C\_1\). This
shows that, on Mackie's theory, an effect of a common cause may
falsely count as a cause for another effect of said common cause.



For Mackie (1974: 85-86), this type of problem indicates that
his regularity theory is at best incomplete. And he tells us what is
missing: a genuine cause-effect relation is marked by causal priority.
As an approximation, a token event \(c\) was causally prior to another
token event \(e\) if an agent could have--in
principle--prevented \(e\) by "(directly or indirectly)
preventing, or failing to bring about or to do \(c\)" (1974:
190). To prevent the sounding of the Manchester hooters does not
prevent the workers from going home in London. Hence, the instance of
\(E\_1\) was not causally prior to the associated instance of \(E\_2\)
(though it is temporally prior). By contrast, preventing the sounding
of the London hooters does prevent the workers from going home in
London.



The approximate relation of causal priority is interventionist. Causes
are seen as devices for an (ideal) agent to manipulate an effect.
However, Mackie did not propose an interventionist theory of
causation. His final notion of causal priority is independent of a
notion of agency. \(c\) was causally prior to \(e\) if there was a
time at which \(c\) was fixed while \(e\) was unfixed (see 1974:
180-183 and 189-192). It should be noted, however, that Mackie himself
does not think that his analysis of causal priority succeeds (1974:
xiv).



Another problem for Mackie's regularity theory is that the
complex regularities seem to be symmetric. This symmetry blurs the
asymmetry between cause and effect, and leads to the problem of the
direction of causation. Like Hume, one could stipulate that causes
precede their effects in time. The direction of causation is thus
simply the direction of time. However, this stipulation excludes both
the possibility of (a) backwards causation and (b) an analysis of
"time's arrow" in terms of causation. Since--to
the best of our knowledge--there is no account of causation which
has fully solved the problem of the causal direction, the objections
(a) and (b) do not speak decisively against the regularity theory.


### 1.4 Contemporary Regularity Theories



Mackie's theory of causation continues to be influential.
Richard Wright (1985, 2011) proposed a similar account for identifying
causes which can be, and indeed is, used in legal contexts. Building
on Hart and Honore (1985), he defined a cause as follows: a
condition \(c\) was a cause of \(e\) iff \(c\) was necessary for the
causal sufficiency of a set of existing conditions that was jointly
sufficient for the occurrence of \(e\). Causal sufficiency is here
different from lawful sufficiency: the set of conditions must
instantiate the antecedent of a causal law whose consequent is then
instantiated by the effect. Wright's account requires that we
have causal laws--as opposed to laws--at our disposal and so
it is only reductive if we have a reduction of the causal laws.
Nevertheless, the idea that a cause is a necessary element of a
sufficient set for an effect has been applied in legal cases to
determine which event has caused another. In fact, this NESS test for
causation has itself become influential in legal theory (see the entry on
 causation in the law).



Strevens (2007) develops Mackie's theory further. On the way, he
likewise gives up its reductive character by replacing sufficiency
with causal sufficiency. The latter is to be understood in terms of
causal entailment, which in turn is supposed to represent a causal
process. (We explain Strevens's theory of causation in greater
detail in
 SS2.4.)
 Notably, Strevens (2007) argues that Mackie's original theory
already solves not only the causal scenario known as
overdetermination--as everyone agrees--but also (early and
late) preemption. (For an overview of problematic causal scenarios,
see Paul & Hall 2013).



Baumgartner (2008, 2013) developed Mackie's theory further while
keeping its reductive character. He observes that complex regularities
like

 

\[(C\_1 \land C\_2) \lor D\_1 \leftrightarrow E\]


are not symmetric in the following sense: an instantiation of a
cluster is sufficient for \(E\), but an instantiation of \(E\) is
generally not sufficient to determine which cluster is instantiated.
An instantiation of \(E\) does not determine whether \(C\_1 \land C\_2\)
or \(D\_1\) is instantiated. An instantiation of \(E\) only determines
the whole disjunction of minimally sufficient clusters. This
non-symmetry may be exploited to establish a notion of causal priority
or the direction of causation.



There is, of course, a problem when a single cluster is necessary and
sufficient for an effect:

 

\[(C\_1 \land C\_2) \leftrightarrow E.\]


Here the instantiation of \(E\) is sufficient to determine the
instantiation of the cluster. Baumgartner's suggestion requires
an effect to have at least two alternative sufficient clusters in
order to establish the direction of causation. And even then, it seems
that the problem of joint effects of a common cause is still unsolved.
Consider
 Figure 2,
 where \(A\) and \(B\) are the joint effects of a common cause \(C\),
and \(D\) and \(E\) are alternative causes for \(A\) and \(B\),
respectively.



![a diagram of 5 disjoint circles labeled A, B, C, D, and E. Arrows go from D to A, C to A, C to B, and E to B.](figure2.svg)

Figure 2




Whenever \(A \land \neg D\) is actual, so is \(C\)--for no effect
occurs without any of its causes. Furthermore, whenever \(C\) occurs,
so does \(B\). Hence, \(A \land \neg D\) is minimally sufficient for
\(C\), and thus for \(B\). And \(A \land \neg D\) is a minimally
sufficient cluster in a necessary condition for \(B\):


\[\tag{1} (A \land \neg D) \lor C \lor E \leftrightarrow B.\]

But \(A \land \neg D\) should not count as a cause of \(B\). After
all, the causes of \(B\) are only \(C\) and \(E\) in this causal
scenario.



Baumgartner proposes a solution. The complex regularity for an effect
must be necessary for that effect in a minimal way. \(B\) is
instantiated only if \(C\) or \(E\) is instantiated. Hence, \(C \lor
E\) is necessary for \(B\). And the left side of the complex
regularity
 (1)
 contains no other disjunction--obtained by removing one
disjunct--which is necessary for \(B\). There are only two
candidates. \((A \land \neg D) \lor C\) is not necessary because \(B\)
can be instantiated along with \(E, \neg C\), and \(A \land D\).
Similarly, \((A \land \neg D) \lor E\) is not necessary because \(B\)
can be instantiated along with \(C,\) \(\neg E\) and \(A \land D\).
Hence, \(B \rightarrow C \lor E\) and no disjunct can be removed from
\(C \lor E\) such that the implication still holds. In this sense, \(C
\lor E\) is a *minimally necessary* disjunction of minimally
sufficient clusters for \(B\).



The notion of a minimally necessary disjunction can also be
characterized in terms of more formal concepts. Suppose \(\mathbf{C}\)
is a set of clusters, where each cluster is given by a conjunction of
factors. \(\mathbf{C}\) is a minimally necessary condition for \(E\)
iff



(i)
\(\bigvee \mathbf{C} \leftrightarrow E\) is true in the actual
world, and
(ii)
there is no \(\mathbf{C}'\subset \mathbf{C}\) such that \(\bigvee
\mathbf{C}' \leftrightarrow E\) is also true in the actual world.


(\(\bigvee \mathbf{C}\) designates some disjunction of the members of
\(\mathbf{C}\).)



The underlying reason why \(A \land \neg D\) can be pruned from the
left side of
 (1)
 is that \(A \land \neg D\) is sufficient for \(C \lor E\), while the
latter is not sufficient for the former in the given scenario. Now, an
instantiation of \(C \lor E\) makes a difference as to whether or not
\(B\) is instantiated--independently of \(A \land \neg D\). The
converse is not true: an instantiation of \(A \land \neg D\) makes a
difference as to whether or not \(B\) is instantiated, but not
independently of \(C \lor E\) (see Baumgartner 2008: 340-346 and
2013: 90-96). This suggests the following requirement: a complex
regularity for an effect must be a minimally necessary disjunction of
minimally sufficient clusters for said effect. Indeed, given that each
effect has at least two alternative causes, this seems to solve the
problem of joint effects of a common cause.



Baumgartner (2013: 95) defines a cause on the type level as follows.
Relative to a factor set containing \(C\) and \(E\), \(C\) is a
type-level cause of \(E\) iff



(i)
\(C\) is a factor of a minimally necessary disjunction of
minimally sufficient clusters for \(E\), and
(ii)
\(C\) remains such a factor for any suitable expansion of the
factor set.


A type-level cause is thus permanently non-redundant for its effect.
He then goes on to define a cause on the token level (for details, see
Baumgartner 2013: 98). Relative to a factor set containing \(C\) and
\(E\), an instance \(c\) of \(C\) is an actual cause of an instance
\(e\) of \(E\) iff \(C\) is a type-level cause of \(E\), and, for all
suitable expansions of the factor set including the original factor
set, \(C\) is on an active causal route to \(E\). An active causal
route is roughly a sequence of factors \(\langle Z\_1, \ldots , Z\_n
\rangle\), where each element \(Z\_i\) is a factor of a minimally
necessary disjunction of minimally sufficient clusters for its
successor element \(Z\_{i+1}\), and each \(Z\_i\) is co-instantiated
with a set of factors \(X\_i\) such that \(Z\_i \land X\_i\) form a
minimally sufficient cluster for \(Z\_{i+1}\). This amounts to a quite
powerful analysis of actual causation. It solves many scenarios which
are troublesome for counterfactual accounts, including
overdetermination, early and late preemption, and scenarios known as
"switches", and "short-circuits". For a
detailed study of these scenarios and an explanation why they are
trouble for the counterfactual approach to causation, see Paul and
Hall (2013). Finally, it should be noted that Baumgartner's
regularity theory faces a problem relating to what Baumgartner and
Falk (forthcoming) call structural redundancies. The latter explain
this problem and offer a solution based on a causal uniqueness
assumption.


## 2. Inferential Theories of Causation



Inferential theories of causation are driven by the idea that causal
relations can be characterized by inferential relations. In essence,
\(C\) is a cause of \(E\) iff



(i)
\(C\) and \(E\) are actual events, and
(ii)
\(E\) can be inferred from \(C\) in the context of a suitable
background theory using a suitable logic.


Inferential theories may be seen as refinements and generalizations of
regularity theories. For example, one can explain INUS conditions in
terms of relatively simple inferences between the presumed cause and
the putative effect. In this explanation, the distinctive feature of
Mackie's theory is explained by the logical form of the
background theory. In a propositional setting, this background theory
has the logical form of a biconditional

 

\[(C\_{1,1} \wedge \ldots \wedge C\_{1,n}) \vee \ldots \vee (C\_{k,1} \wedge C\_{k,m}) \leftrightarrow E.\]


Each \(C\_{i,j}\) that co-occurs with all conjuncts of one conjunction
on the left-hand side of this biconditional is at least an INUS
condition for the effect \(E\). Obviously, we can infer \(E\) from
such a \(C\_{i,j}\) in the context of the biconditional and the other
conjuncts that form, together with \(C\_{i,j}\), a sufficient condition
for \(E\). However, the exposition of the INUS account in Mackie
(1965, 1974) is not explicitly inferential. Likewise, Hume's
original regularity theory can be interpreted inferentially, but it is
not an explicitly logical or inferential account.



We understand the notion of an inference relation in a broad way. In
essence, an inference relation maps sets of sentences to sets of
sentences such that this mapping is guided by some idea of truth
preservation: if all members of a given set of premises are true, then
all sentences inferable from this set must be true as well. Moreover,
there are inference relations that are guided by a weaker requirement:
if all members of a given set of premises are believed, then it is
rational to believe a certain set of conclusions. Nonmononotonic
logics and formal theories of belief revision define inference
relations in this sense. We shall say more about such inference
systems in
 Section 2.2.
 Suffice it to say for now that an inference relation understood in
our broad way may be defined syntactically in terms of relations of
derivability, or semantically in terms of possible worlds and
model-theoretic interpretations, or both ways.



Inferential theories seek to improve on regularity theories. Recall
from
 Section 1.1
 that virtually all regularity theories face two problems of
extension. First, in the case of singular causal relations, we have a
causal relation without regularity. Second, in the case of spurious
causation, we have a regularity that is not considered to be causal. A
sophisticated inferential theory may allow us to infer singular
events, such as the nuclear disaster in Fukushima and the extinction
of the dinosaurs, from a rich background theory and certain further
hypotheses about presumed causes and the context of the presumed
causal process. At least some inferential theories hold the promise to
distinguish between spurious and genuine causal relations. The idea is
that genuine causal relations may be characterized by distinctive
inferential relations, and so be distinguished from spurious causal
relations.


### 2.1 Deductive Nomological Approaches



Inherent in the philosophy of logical empiricism, broadly construed,
is a strong opposition to traditional metaphysics. Not surprisingly,
many of the logical empiricists were guided by Humean empiricist
principles. Even some concepts of scientific language were suspected
to be metaphysical in nature. Wittgenstein (1922: 5.136) held that
belief in a general law of causality is superstition. He also denied
that laws of nature could explain natural phenomena (6.371n). Frank
(1932: Ch. V, IX) pointed out several difficulties in giving a precise
formulation of a general law of causality. Russell (1913) argued that
the notion of cause is beset with confusion, and should better be
excluded from philosophical vocabulary.



Ramsey (1931) was less sceptical than Russell (1913) and Wittgenstein
(1922) about the prospect of an analysis of causation. His account of
causation seems to be the logical empiricist analysis that comes
closest to Hume's original analysis. It is centred on the notion
of a *variable hypothetical*. Such a hypothetical is a
universally quantified implication that has instances whose truth or
falsity we are unable to verify for now. A case in point is the
sentence saying that all humans are mortal. This sentence goes beyond
our finite experience. For there is no way to conclusively establish
that all men--past, present and future--are mortal.



To believe a universal hypothetical consists in "a general
enunciation" and "a habit of singular belief"
(Ramsey 1931: 241). That is, we are willing to believe instances of
the universal sentence, even though we are not able to verify them at
the moment. Ramsey builds here Humean ideas about the formation of a
habit or custom into an informal semantics of universal sentences.



Ramsey's account, then, amounts to the following analysis. \(C\)
is a cause of \(E\) iff



(i)
\(C\) and \(E\) are actual,
(ii)
the implication \(C \rightarrow E\) follows from a variable
hypothetical \(\forall x (\phi(x) \rightarrow \psi(x))\) and certain
facts \(F\),
(iii)
the events described by \(E\) are not earlier than the events
described by \(C\) and \(F\),
(iv)
\((C \wedge F) \rightarrow E\) is an instance of \(\forall x
(\phi(x) \rightarrow \psi(x))\), and
(v)
\(\forall x (\phi(x) \rightarrow \psi(x))\) is believed, or we are
led to believe it in light of future observations (Ramsey 1931:
249).


Condition (i) remains implicit, but it would be confusing to omit it
here. Strictly speaking, condition (ii) is redundant since it follows
from condition (iv).



Note that a regularity by itself need not give rise to a variable
hypothetical that we believe or are led to believe by future
observations. A case in point is the regular connection between the
sound of the hooters at a factory in Manchester and the workers going
home at a factory in London (see
 SS1.3).
 Ramsey (1931: 242) literally speaks of trust in order to characterize
our epistemic attitude towards causal laws. Ramsey's account of
such laws, which supersedes his earlier best systems analysis, is
explicitly epistemic. However, Ramsey's account does not seem to
have the resources to address the problem of singular causal
relations. Under which causal law could we subsume the causal
hypothesis that dinosaurs became extinct by a collision of the Earth
with a meteor?



More work on causation and explanation has been done within the
logical empiricist program. The deductive nomological model of
explanation by Hempel and Oppenheim (1948)--also referred to as
the DN model of explanation--seems to give us an exemplar of a
conceptual analysis without metaphysics. Let us briefly recall the
basic elements of this model. The explanans of an empirical phenomenon
\(E\) consists of two types of statements. First, a set \(C\) of
antecedent conditions \(C\_1, \ldots, C\_k\). Second, a set \(L\) of
laws \(L\_1, \ldots, L\_r\). These two sets form the *explanans*,
while \(E\) is called the *explanandum*. \(C\) and \(L\)
explain \(E\) iff



(i)
\(E\) is a logical consequence of the union of \(C\) and
\(L\),
(ii)
\(L\) is non-empty (while \(C\) may be empty),
(iii)
the explanans is testable, and
(iv)
each member of \(C\) and \(L\) is true.


The explanandum must thereby be subsumed under at least one law.



While Hempel and Oppenheim were not primarily interested in an
analysis of causation, they thought that the DN model may yield such
an analysis:



> 
> 
> The type of explanation which has been considered here so far is often
> referred to as causal explanation. If \(E\) describes a particular
> event, then the antecedent circumstances described in the sentences
> \(C\_1, C\_2, \ldots, C\_k\) may be said jointly to "cause"
> that event, in the sense that there are certain empirical
> regularities, expressed by the laws \(L\_1, L\_2, \ldots, L\_r\), which
> imply that whenever conditions of the kind indicated by \(C\_1, C\_2,
> \ldots, C\_k\) occur, an event of the type described in \(E\) will take
> place. Statements such as \(L\_1, L\_2, \ldots, L\_r\), which assert
> general and unexceptional connections between specified
> characteristics of events, are customarily called causal, or
> deterministic, laws. (Hempel & Oppenheim 1948: 139)
> 
> 
> 



This passage nicely indicates how inferential approaches to causation
parallel regularity approaches. The regular connection between the
presumed cause and the putative effect is expressed by certain laws.
The putative effect must be inferable from the presumed cause using
these laws if the former is an actual cause of the latter. Causation
implies that there are covering laws, according to which the cause is
sufficient for its effect (cf. Pap 1952 and Davidson 1967, 1995). At
the same time, the above passage reveals logical empiricist scruples
to talk about causation. Hempel and Oppenheim are hesitant to set
forth the DN model as an analysis of causation and to say that the
antecedent conditions of a DN explanation are causes of the
explanandum if the latter is a particular event.



More than a decade later than the first publication of the DN model,
Hempel (1965: 351n) suggested replacing the notion of cause by the
notion of antecedent conditions along the lines of the DN model.
Moreover, he then distinguishes more carefully between *laws of
succession* and *laws of coexistence* (1965: 352). Only the
former give rise to causal explanations. If a DN explanation is based
on laws of coexistence, this explanation does not qualify as causal. A
case in point is the law that connects the length of a pendulum with
its period:

 

\[T=2 \pi \sqrt{l/g}\]


where \(T\) is the period of the pendulum, \(l\) its length, and \(g\)
the gravitational acceleration nearby the surface of the Earth.



Arguably, even laws of succession do not always give rise to causal
explanations. Such laws do not only allow for predictions, but also
for retrodictions. That is, we can derive statements about events that
are prior to events described by the antecedent conditions \(C\_1,\)
\(C\_2,\) ..., \(C\_k\) in a DN explanation. Hempel (1965: 353)
himself observed that Fermat's principle may be used in a DN
explanation of an event that precedes an event described by an
antecedent condition.



In sum, there are at least two problems with a simple DN account of
causation. First, DN explanations using laws of coexistence do not
seem to qualify as causal. Second, in some DN explanations, the
explanandum is an event that occurs prior to an event described in the
antecedent conditions. In neither case, we view the antecedent
conditions of the DN explanation as cause of the event to be
explained.



These observations anticipate later criticisms of the DN model, which
are commonly referred to as *symmetry problems*. In the case of
the infamous tower-shadow example, most people have come to view only
the derivation of the length of the shadow as properly explanatory,
but not the derivation of the height of the tower (Bromberger 1966).
Hempel (1965: 353n) himself admits that causal DN explanations seem
"more natural and plausible" than non-causal ones.
Woodward (2003) goes a step further by expounding a causal account of
explanation using causal models.



We must wonder why Hempel did not define that \(C\) is a cause of a
particular event \(E\) iff there is a DN explanation of \(E\) such
that \(E\) is the explanandum, \(C\) is among the antecedent
conditions, all antecedent conditions precede the occurrence of \(E\),
and all laws are laws of succession. Presumably, his interest in an
analysis of causation was limited.


### 2.2 Ranking Functions



Spohn (2006, 2012) expounds a broadly Humean analysis of causation.
\(C\) is a cause of \(E\) iff



(i)
\(C\) and \(E\) occur,
(ii)
\(C\) precedes \(E\), and
(iii)
\(C\) "raises the metaphysical or epistemic status" of
\(E\), given the circumstances of the causal scenario (Spohn 2006:
97).


(iii) says that \(C\) is a reason for \(E\), given the circumstances.
This relation is analysed in terms of ranking functions. Spohn views
this analysis as an alternative to counterfactual approaches. At the
end of this section, we will understand why the analysis qualifies as
an inferential approach to causation.



Spohn develops the ranking-theoretic analysis in two steps. First, he
explains what it is for a proposition to be a reason for another
proposition. Second, causal relations are characterized as specific
types of reason relations. Let us go a bit further into details. As
regards notation, we follow Spohn (2006). The notation in Spohn (2012)
is more refined, but slightly more complex as well.



A ranking function represents a belief state. More specifically, a
ranking function \(\kappa\) assigns each possible world a non-negative
natural number such that \(\kappa(w)=0\) for at least one possible
world \(w\). Intuitively, the rank \(\kappa(w)\) of a possible world
\(w\) expresses a degree of disbelief. \(\kappa(w)=0\) means that the
degree of disbelief is zero. \(w\) is not disbelieved in this case.
Otherwise, it is disbelieved with rank \(\kappa(w)=n > 0\). A
ranking function thus defines a Grovian system of spheres, used to
represent AGM belief revision operators (Alchourron,
Gardenfors, & Makinson 1985; Grove 1988). A Grovian system of
spheres is basically a set of nested sets. At the center of the system
is its smallest member, a non-empty set of worlds. The smallest set is
surrounded by its supersets, as shown in
 Figure 3.



![a diagram of 4 concentric circles labeled innermost to outer most as 0, 1, 2, 3. An area, labeled 'not-A' is defined by a boundary curve and it includes a portion of all the circles except circle 0. The intersection of the 'not-A' area and circle 1 is shaded gray.](figure3.svg)

Figure 3




The ranks 0, 1, 2, 3,... are understood as cardinal numbers. For
example, if world \(w\_1\) has rank 5 and world \(w\_2\) rank 10, then
\(w\_2\) is disbelieved to a degree twice as much as \(w\_1\) is
disbelieved. Some ranks may even be empty in the sense that there is
no world that has a specific rank \(n\). For example, ranks 3 and 4
may be without any possible worlds, while there are possible worlds at
ranks 0, 1, 2, and 5. The cardinal interpretation of the ranks is
needed for certain arithmetical operations, such as lifting the ranks
by a certain cardinal number, applied to all possible worlds of a
certain subset. Because of the possibility of empty ranks, a Grovian
system of spheres does not define a unique ranking function.



At each possible world, either a proposition or its negation is true.
We call a world at which a proposition \(A\) is true an \(A\)-world,
and identify the proposition \(A\) with the set of \(A\)-worlds. The
rank \(\kappa(A)\) of a proposition \(A\) is the rank of those
possible worlds \(w\) whose rank \(\kappa(w)\) is minimal among the
\(A\)-worlds. In
 Figure 3,
 the rank of \(A\) is thus \(0\) and the rank of \(\neg A\) is \(1\).
A proposition \(A\) is believed iff \(\kappa(\neg A)>0\).



Ranking functions cannot only be used to express degrees of disbelief,
but also to express degrees of belief. This is accomplished by the
belief function \(\beta\). Where \(A\) is a proposition,



\[\beta(A)= \kappa(\neg A) - \kappa(A).\]


That is, \(\beta(A)\) is the difference between the minimal rank of
the \(\neg A\)-worlds and the minimal rank of the \(A\)-worlds. For
example, if \(\kappa(\neg A)\) is rather high, we strongly believe
\(A\). \(\beta(A)>0\) means that \(A\) is believed to be true, and
\(\beta(A)<0\) that \(A\) is believed to be false. \(\beta(A)=0\)
means that \(A\) is neither believed nor disbelieved. In the latter
case, we have both \(A\)- and \(\neg A\)-worlds whose rank is
zero.



We are almost done with the core of ranking theory. It remains to
explain *conditionalization*. Conditionalization of a ranking
function on a proposition represents the belief change upon coming to
believe the proposition. The conditionalization of \(\kappa\) on \(A\)
is defined as follows:

 

\[\kappa(w\mid A)= \kappa(w) - \kappa(A)\]


for all worlds \(w\) at which \(A\) is true, while \(\kappa(w\mid A)=
\infty\) for all worlds \(w\) where \(A\) is false. In less formal
terms, if our agent comes to believe \(A,\) then the ranks of all
\(A\)-worlds are lowered by the rank of \(A\). And the rank of all
\(\neg A\)-worlds is set to infinity, which means that she definitely
disbelieves those worlds. The conditionalization of a belief function
\(\beta\) is defined analogously:

 

\[\beta(B\mid A) = \kappa(\neg B \: |\: A) - \kappa(B \: |\: A).\]


These ranking-theoretic concepts at hand, Spohn defines what it is for
a proposition to be a reason for another proposition. \(A\) is a
reason for \(B\) in the context \(C\)--relative to
\(\beta\)--iff

 

\[\beta(B \: |\: A \cap C) > \beta(B \: |\: \neg A \cap C).\]


Moreover, he introduces subtle distinctions between an additional, a
sufficient, a necessary, and a weak reason. These distinctions give
rise to the notions of an additional, a sufficient, a necessary, and a
weak cause. We leave out these subtleties for simplicity.



Now that we know what a ranking-theoretic reason is, it remains to
explain which types of reasons are causes. In essence, \(C\) is a
*direct cause* of \(E\) at the possible world
\(w\)--relative to the belief function \(\beta\)--iff



(i)
\(C\) and \(E\) are true at \(w\),
(ii)
the event expressed by \(C\) precedes the event expressed by
\(E\), and
(iii)
\(C\) is a reason for \(E\) in the context \(K\), which is given
by the past of \(E\), excluding information about \(C\).


Spohn thinks that we should understand causation relative to *small
worlds* such that the context is given in a certain *frame*
of possible events, but not by the global past of \(E\). Causation is
thus doubly relative to an epistemic perspective. First, it is
relative to a belief function, and second to a frame of possible
events.



A final step is needed to also capture non-direct causal relations.
\(C\) is a cause of \(E,\) possibly non-direct, iff the ordered pair
\((C,E)\) is in the transitive closure of the relation of direct
causation. That is, \(C\) is a cause of \(E\) iff there is a sequence
\(\langle C, C\_1, \ldots, C\_n, E \rangle\) \((n\geq 0)\) such that
each element of the sequence is a direct cause of the successor (if
there is one).



Most notably, the ranking-theoretic account of causation is able to
solve the problem of spurious causation, arising from common cause
scenarios. The solution is quite subtle. Suppose \(C\) is a common
cause of \(E\) and \(F,\) where \(F\) precedes \(E\). The crucial
question then is whether or not \(F\) is a reason for \(E\) in the
context \(C\). Spohn (2006: 106) argues that it is not. Recall that
\(F\) is a reason for \(E\) in the context \(C\) iff

 

\[\beta(E\mid C \cap F) > \beta(E\mid C \cap \neg F),\]


where

 

\[\beta(E\mid C \cap F)=\kappa(\neg E\mid C \cap F) - \kappa( E\mid C \cap F),\]


and

 

\[\beta(E\mid C \cap \neg F)=\kappa(\neg E\mid C \cap \neg F) - \kappa( E\mid C \cap \neg F).\]


Spohn determines the difference between the ranks \(\kappa(\neg E \mid
C \cap F)\) and \(\kappa( E\mid C \cap F)\) by the difference in the
number of violations of causal regularities between the world where
\(\neg E \land C \land F\) is true and the world where \(E \land C
\land F\) is true. In the former world, the causal regularity between
\(C\) and \(E\) is violated, while the causal regularity between \(C\)
and \(F\) is respected. No causal regularity is violated in the world
where \(E \land C \land F\) is true. Hence,

 

\[\beta(E\mid C \cap F)=1.\]


As for the calculation of \(\beta(E\mid C \cap \neg F),\) we have two
violations of a causal regularity in the world of \(\neg E \land C
\land \neg F\), but only one such violation in the world of \(E \land
C \land \neg F\). Hence,

 

\[\beta(E\mid C \cap \neg F)=1.\]


It therefore holds that

 

\[\beta(E\mid C \cap F) = \beta(E\mid C \cap \neg F).\]


Hence, \(F\) is neither a reason nor a cause of \(E\). It is easy to
show that \(C\) is a reason for \(E\), as desired.



Spohn (2006, 2012) considers a number of further causal scenarios, and
so he offers ranking-theoretic solutions to the problems of
overdetermination, trumping, and various scenarios of preemption. The
account of overdetermination parallels probabilistic accounts of this
type of causal scenario. Spohn does not explicitly address singular
causal relations that lack a corresponding regularity. But there might
be a way to argue that a collision of the Earth with a huge
meteor--at the time when dinosaurs were roaming the
Earth--raises the ranks of those worlds where dinosaurs are
alive. Such a raise amounts to believing that dinosaurs could have
died because of such a collision.



Why is Spohn's analysis of causation inferential? Recall that a
ranking function represents the epistemic state of an agent, which
implies that it determines which beliefs are held by the agent.
Moreover, the conditionalization of a ranking function defines how
such a function changes if the agent comes to believe a certain
proposition. Hence, conditionalization defines an inference relation
that maps a set of propositions to a powerset of propositions. The
intended meaning of this inference relation is that, if we come to
believe a certain proposition, then it is rational to believe a set of
propositions. In other words, proposition \(B\) is inferable from
proposition \(A\) in the context of a ranking function \(\kappa\) iff
\(\beta(B\mid A)>0\), where \(\kappa\) is underlying the belief
function \(\beta\). Put into symbolic notation:

 

\[A \infd\_\kappa B \;\text{ iff }\; \beta(B\mid A)>0.\]


This notation is not used by Spohn (2012), but may be helpful to grasp
the inferential nature of Spohn's condition that a cause must be
a reason for its effect. Notice that \(\infd\_\kappa\) is an
enthymematic inference relation in the sense that certain premises are
hidden in the background.



The inferential nature of the condition that a cause must be a
ranking-theoretic reason for its effect can also be seen from the
following two results. First, the conditionalization of a belief
function \(\beta\) defines a belief revision scheme (Spohn 1988). Such
a scheme tells us how a rational agent should change his or her
beliefs in light of new information (Gardenfors 1988). Second,
any belief revision scheme--in the sense of the AGM theory by
Alchourron, Gardenfors, and Makinson (1985)--defines
a nonmonotonic and enthymematic inference relation (Gardenfors
1991). This result motivates the above definition of the inference
relation \(A \infd\_\kappa B\).



Spohn's theory of causation is first expounded in terms of
ranking functions which model the epistemic states of some agents.
However, it should be noted that Spohn (2012: Ch. 14) aims to spell
out the objective counterpart to ranking functions. He seeks to
*objectify* ranking functions by saying what it means that a
ranking function represents objective features of the world. The
notion of truth may serve as an introductory example. Given \(w\) is
the actual world, a ranking function \(\kappa\) is an objective
representation of this world iff \(\beta(A)>0\) for all
propositions \(A\) such that \(w\in A\). (Proposition \(A\) is
construed as a set of possible worlds; \(\beta\) is the belief
function of \(\kappa\), as defined above.) This explanation, however,
does not yet capture modal and causal features of our world. And so
Spohn goes on to explain how modal and causal properties--defined
by a given ranking function \(\kappa\)--can be understood as
objectives features of the world. The details are intricate and
complex. They are spelled out in (2012: Ch. 15); the basic ideas are
summarized in Spohn (2018). Suffice it to say that Spohn's
objectification of ranking functions is not only a promising attempt
for an objective account of causation and modality, but also a way to
resolve the tension between an epistemic and non-epistemic theory of
causation.


### 2.3 Strengthened Ramsey Test



Drawing on Spohn (2006), Andreas and Gunther (2020) analyse
causal relations in terms of certain reason relations. This is the
basic schema of the analysis: \(C\) is a cause of \(E\) iff



(i)
\(C\) is a reason for believing \(E\), and
(ii)
the ordered pair \((C, E)\) satisfies a few further
conditions.


Andreas and Gunther specify the first condition by a strengthened
Ramsey Test, which emerges from an analysis of the word
"because" in natural language. Ramsey (1931) proposed to
evaluate conditionals in terms of belief change. Roughly, you accept
"if \(A\) then \(C\)" when you believe \(C\) upon
supposing \(A\). This Ramsey Test has been pointedly expressed by
Stalnaker:



> 
> 
> First, add the antecedent (hypothetically) to your stock of beliefs;
> second, make whatever adjustments are required to maintain consistency
> (without modifying the hypothetical belief in the antecedent);
> finally, consider whether or not the consequent is then true. (1968:
> 102)
> 
> 
> 



Gardenfors (1988) expressed the Ramsey Test in more formal terms.
Where \(K\) denotes an agent's set of beliefs and \(K \ast A\)
the operation of changing \(K\) on the supposition of \(A,\) \(A >
C \in K\) if and only if \(C \in K \ast A.\) In the wake of work by
Rott (1986), Andreas and Gunther suggest strengthening the Ramsey
Test as follows:



> 
> 
> \(A \gg C\) iff, after suspending judgment about \(A\) and \(C\), an
> agent can infer \(C\) from the supposition of \(A\) (in the context of
> further beliefs in the background). (2019: 1230)
> 
> 
> 



This conditional is then used to give a simple analysis of the word
"because" in natural language:

 

\[\text{Because } A, C \text{ (relative to }K)\;\;\;\text{ iff }\;\;\;A \gg C \in K \;\;\; \text{ and }\;\;\; A, C \in K\]


where \(K\) designates the belief set of the agent. This analysis may
well be incomplete, but proves useful for analysing causation. The
next step toward such an analysis is to impose further constraints on
the inferential relations between the antecedent and the consequent.
For the expert reader, it may be worth noting that the conditional
\(\gg\) is defined in terms of finite belief bases rather than
logically closed sets. Inferential relations between antecedent and
consequent can therefore be defined syntactically in terms of
derivability.



How may the conditional \(\gg\) be used to analyse causation? Suppose
the antecedent \(C\) designates a presumed cause and the consequent
\(E\) a putative effect. It is first required by Andreas and
Gunther (2020) that we can infer \(E\) from \(C\) in a
forward-directed manner. That is, there is an inferential path between
\(C\) and \(E\) such that, for every inferential step, whenever we
infer the occurrence of an event, no premise asserts the occurrence of
an event that is temporally later than the inferred event. This
condition is a refined and slightly modified variant of Hume's
requirement according to which a cause always precedes its effect.



Second, the inferential path between \(C\) and \(E\) must be such that
every intermediate conclusion is consistent with our beliefs about the
actual world. The idea is that there must be an inferential path
between the presumed cause and the putative effect that tells us how
the effect is brought about by the cause. This inferential
representation of how the effect is produced by the cause must be
consistent with what we believe the world is like.



Third, every generalization--used in the inferential path between
\(C\) and \(E\)--must be non-redundant in the following sense: it
must not be possible to derive the generalization from other explicit
beliefs. (A generalization is simply a universal sentence, which may
represent a strict or non-strict law.) By means of this condition,
Andreas and Gunther (2020) try to solve the problem of spurious
causation in common cause scenarios. Suppose there is a thunderstorm.
Then, we have an electrical discharge that causes a flash which is
followed by thunder. The flash precedes the thunder, and there is a
regular connection between these two types of events. But we would not
say that the flash is a cause of the thunder. So, we need to
discriminate between forward-directed inferences with a causal meaning
and other forward-directed inferences that lack such a meaning. Note
that we can derive the event of the flash and the event of the thunder
from the electrical discharge in a forward-directed manner using only
non-redundant generalizations, that is, universal sentences that we
think are relatively fundamental laws of nature. Among these are the
laws of electrodynamics, atomic and acoustic theory, and optics.
However, the only inferential path from the flash to the thunder that
is forward-directed requires redundant generalizations: universal
sentences that we can derive from more fundamental laws (see Andreas
2019 for details). The strategy is that spurious causal relations can
be excluded by requiring the generalizations used in the inferential
path to be non-redundant. This strategy works in the mentioned common
cause scenario. It remains to be seen whether this strategy works in
full generality.



In sum, \(C\) is a cause of \(E\)--relative to a belief
state--iff



(i)
\(C\) and \(E\) are believed to be true,
(ii)
the event of \(C\) precedes that of \(E\), and
(iii)
after suspending judgement about \(C\) and \(E\), we can infer
\(E\) from \(C\) in a forward-directed manner, without using redundant
generalizations, and without inferring sentences that are inconsistent
with our beliefs.


Obviously, this analysis is epistemic, just as Hume's analysis
of causation in the mind as well as Ramsey's and Spohn's
analyses are. Condition (ii) is needed because this condition is not
implied by the requirement that the inferential path between \(C\) and
\(E\) is forward-directed. (Such an implication does not hold because,
strictly speaking, forward-directedness of the inferential path means
that no inferential step is backward-directed.) The problem of
instantaneous causation is addressed in Andreas (2019), at least for
some causal scenarios. Andreas and Gunther (2020) deliver
rigorous solutions to a number of causal scenarios, including
overdetermination, early and late preemption, switches, and some
scenarios of prevention. Singular causal relations without a regular
connection among \(C\)-type and \(E\)-type events are not
addressed.


### 2.4 The Kairetic Account



The kairetic account by Strevens (2004, 2008) analyses causation in
terms of causal models. It is driven by the idea that every causal
claim is grounded in a causal-explanatory claim. Causation and
explanation go hand in hand. Strevens uses the notion of a causal
model to define a relation of entailment with a causal meaning. A set
of propositions entail an explanandum \(E\) in a causal model only if
this entailment corresponds to a "real causal process by which
\(E\) is causally produced" (Strevens 2004: 165). Causal models
are assumed to be founded in physical facts about causal influence,
and it is assumed that these facts "can be read off the true
theory of everything" (Strevens 2004: 165).



The kairetic account parallels the DN account of causation in some
respects. The logical form of the explanans is basically the same as
in the DN model. We have laws and propositions about events. A causal
model is here simply a set of causal laws taken together with a set of
propositions about potential causal factors. Unlike the DN account of
causation, the kairetic analysis uses causal notions in the analysans.
More specifically, the notion of entailment with a causal meaning is
taken as antecedently given. In the previous section, we have
mentioned Andreas's (2019) attempt to further specify the
relation of logical entailment with a causal meaning using non-modal
first-order logic, belief revision theory, and some further
non-logical concepts. This specification may be taken to supplement
the kairetic account.



In Strevens (2004), the kairetic account is essentially motivated by
three types of problems. First, causation by conditions that are most
of the time but not always sufficient to produce a corresponding
effect. Second, problems of preemption. Third, the problem of how
fine-grained the description of a cause should be. Consideration of
these problems leads to a sophisticated and powerful analysis of
causation, which is based on the notion of an *explanatory
kernel* for an event. Such a kernel contains laws and propositions
about events. Given the veridical causal model \(M\), the explanatory
kernel for an event \(E\) is the causal model \(K\) such that (i)
\(M\) entails \(K\), (ii) \(K\) entails \(E\), and (iii) \(K\)
optimizes two further properties called *generality* and
*cohesion*. Generality means that \(K\) encompasses as many
physical systems as can be. Cohesion is a measure for the degree to
which potential causal factors are active in all systems satisfying
\(K\) (Strevens 2004: 171). Eventually, Strevens defines \(C\) to be a
causal factor for \(E\) iff it is a member in some explanatory kernel
for \(E\). Causes, in the kairetic account, are difference-makers with
respect to the effect. If we were to take an actual cause out of an
explanatory kernel for \(E\), the remainder of the kernel would not
produce the putative effect \(E\).



Let us try to understand why the description of the cause should be
maximally general. There are three reasons for this. First, this
requirement allows us to eliminate potential causal factors that do
not contribute to the causal production of the effect. A kernel \(K\)
that does mention such a factor--in addition to the
"active" causal factors--is less general than a
kernel \(K'\) that mentions only the "active" causal
factors. Hence, \(K'\) should be preferred over \(K\), other things
being equal.



Put simply, some elements of a veridical causal model are not involved
in the production of the effect. Other elements are involved, but
their influence can be neglected, provided the description of the
effect is not overly fine-grained. For example, the gravitational
forces of other planets may influence the trajectory of a rock thrown
nearby the Earth's surface. But these forces do not make a
difference to whether or not the rock hits a fragile object. Hence,
the kernel should not contain such forces. More generally, a kernel
for \(E\) should not contain elements of \(M\) that are involved in
the production of \(E\) without actually making a difference to it.
This gives us a second reason for the requirement of generality.



These two reasons for optimizing generality are motivated by a
consideration of difference-making: each member of the kernel should
make a difference to the production of the effect. If some member of
the kernel is absent, the remainder of the kernel does not suffice to
bring about the effect. Otherwise, it is not a proper kernel for the
effect.



The third reason for optimizing generality of the kernel is a bit more
intricate. Suppose a brick of four kilograms is thrown against a
window such that the window shatters. Then, for this effect to be
brought about, it is often not necessary that the brick has a weight
of exactly four kilograms. A brick that is slightly lighter and thrown
with the same velocity would also do. Nor does it have to be a brick,
as we know from the rock-throwing examples. So it seems more accurate
to say that the shattering of the window was caused by the fact or
event that a rigid, non-elastic, and sharp object was thrown against
the window in such a manner that the object's impact was within
a certain interval. The latter description is obviously more general
than the former in the sense that more physical systems satisfy it. In
brief, the kernel for \(E\) should not contain aspects of causally
relevant factors that do not make a difference to \(E\).



The requirement of cohesion is needed to exclude disjunctions of
causal factors from a kernel. Suppose \(C\_1\) is a potential, but
inactive causal factor, while \(C\_2\) is active. That is, \(C\_2\) is
actually contributing to the causal production of \(E\), and \(C\_1\)
is not. Then neither \(C\_1\) nor \(C\_1 \vee C\_2\) should be part of
the kernel. We have just seen how \(C\_1\) is excluded from the kernel
by the requirement to maximize generality. \(C\_1 \vee C\_2\) is
excluded by requiring that the kernel should be maximally cohesive.
Other things being equal, kernels with \(\{C\_1 \vee C\_2\}\) and
\(\{C\_1, C\_1 \vee C\_2\}\) are less cohesive than the kernel containing
just \(\{C\_2\}\).



Strevens (2008) is more comprehensive, and addresses a number of
further topics, most of which are primarily concerned with the notion
of an explanation. Notably, an account is offered for singular causal
relations without a corresponding regular connection between the
presumed cause and the putative effect (Ch. 11).


### 2.5 Causal Model Approaches



In his seminal book on causality, Pearl (2000) expounded a formal
framework of causal models, which is centred on structural equations.
A structural equation

 

\[v\_i=f\_i(pa\_i, u\_i),\;\; i=1, \ldots, n\]


tells us that the values \(pa\_i\) of certain parent variables
determine the value \(v\_i\) of a child variable according to a
function \(f\_i\) in the context of a valuation \(u\_i\) of background
variables. At bottom, a causal model is a set of structural equations.
Each equation stands for some causal law or mechanism. We can view
these equations as representing elementary causal dependencies. The
notion of a parent variable relies on causal graphs, which represent
elementary causal relations by directed edges between nodes. Each node
stands for a distinct variable of the causal model. Causal models have
been used to study both deterministic and probabilistic causation.
(For details, see the entry on
 causal models.)



A distinctive feature of the semantics of structural equations is that
it encodes some notion of asymmetric determination. The values of the
parent variables determine the value of the child variable, but not
the other way around. Together with Halpern, Pearl devised several
counterfactual analyses of causation in the framework of causal
models. These analyses incorporate elements of the inferential
approach to causation. To be more precise, condition AC2(a) of the
original and the updated Halpern-Pearl definition of actual causality
is counterfactual, while AC2(b) is inferential. The latter condition
requires that a given effect must be a consequence of its cause for a
range of background conditions. (See Halpern (2016) for details.)
Notably, the framework of causal models has also been used to develop
inferential approaches to causation that work without a counterfactual
condition along the lines of Lewis (1973). We explain the basic
concepts of the accounts by Beckers and Vennekens (2018) and Bochman
(2018).



Beckers and Vennekens (2018) begin with explaining what it is for the
value of a variable to determine the value of another
variable--according to a structural equation--in a context
\(L\). This explanation gives us the notion of \(C\) being a direct
possible contributing cause of \(E\). Such a direct possible
contributing cause is actual if \(C\), \(E\), and the context \(L\)
are actual. For \(C\) to be an actual contributing cause of \(E\),
there must be a sequence

 

\[\langle C, C\_1, \ldots, C\_n, E\rangle\quad (n\geq 1)\]


such that each element of the sequence is a direct actual contributing
cause of its successor (if there is one). Finally, \(C\) is a cause of
\(E\) iff



(i)
\(C\) is an actual contributing cause of \(E\),
(ii)
the causal path between \(C\) and \(E\) satisfies a specific
temporal constraint, while
(iii)
there would be no such path if \(\neg C\) were realized instead of
\(C\).


The temporal constraint basically requires that the events of the
sequence \(\langle C, C\_1, \ldots, C\_n, E\rangle\) are temporally
ordered, i.e., no element in the sequence occurs later than its
successor.



Beckers and Vennekens (2018) deliver rigorous solutions to a number of
causal problems, including overdetermination, early and late
preemption, switches, and some scenarios of prevention. The account is
non-reductive since structural equations represent elementary causal
dependencies. The framework of causal models does not give us a
further analysis of such dependencies. The benefit of this strategy is
that the problem of spurious causal relations in common cause
scenarios does not arise in the first place. The problem of singular
causal relations is not explicitly addressed, but it allows for a
trivial solution in the framework of causal models. We can build a
causal model with a structural equation saying that a collision of the
Earth with a huge meteor leads to the extinction of dinosaurs. A
non-trivial solution is as challenging as in the other inferential
approaches. It seems very difficult to cite all the different causal
laws and background conditions that causally determine dinosaurs to
die.



The analysis by Beckers and Vennekens (2018) is inferential in spirit,
but not explicitly inferential in the sense that they define a logical
inference system with structural equations. Bochman (2018) goes a step
further in this respect. His analysis is centred on what computer
scientists call a *production inference relation*. This
inference relation is characterized by certain metalogical axioms,
such as Strengthening (of the antecedent), Weakening (of the
consequent), Cut, and Or. To get some understanding of a production
inference relation, it suffices to know that it can be reduced to a
set of rules

 

\[A\_1 \wedge \ldots \wedge A\_k \Rightarrow B\_1 \vee \ldots \vee B\_n\]


where \(A\_1, \ldots, A\_k\) and \(B\_1, \ldots, B\_n\) are propositional
literals, i.e., propositional atoms or negations thereof. Such a rule
represents an elementary causal dependency in manner akin to a
structural equation. The symbol \(\Rightarrow\) stands for some notion
of asymmetric determination accordingly. Bochman (2018) shows how
structural equations using only binary variables may be translated
into a set of rules.



It seems as if we could now define \(C\) to be a cause of \(E\) iff we
can infer \(E\) from \(C\) in the context \(K\). But things are more
complicated. The challenge is that the inferential path between \(C\)
and \(E\) must also represent some active causal path, i.e., a
sequence of events such that each event actually caused its successor
(if there is one). In other words, we need some notion of an
inferential path that corresponds to a causal path in the actual
setting of events. Once such a notion is established, Bochman defines
\(C\) to be a cause of \(E\) iff (i) we can infer \(E\) from \(C\) in
the context \(K\), while (ii) we cannot infer \(E\) from the context
\(K\) alone. The details of this inferential account are a bit
intricate, and so we must refer the reader to Bochman (2018) to get a
full understanding. There, we can also find rigorous solutions to a
number of causal problems, including overdetermination, early and late
preemption, and some scenarios of prevention. It is worth noting,
finally, that Bochman presents his analysis as an inferential
explication of NESS conditions.


## 3. Conclusion



Regularity theories of causation have lost acceptance among
philosophers to a considerable extent. One reason is David
Lewis's (1973) influential attack on the regularity approach
which concludes that its prospects are dark. And indeed, the problem
of spurious causation--to pick just one--was severe for the
regularity and inferential theories back then. Mackie's INUS
account and simple deductive-nomological accounts alike cannot
properly distinguish genuine from spurious causes, for example in
common cause scenarios. In the meantime, however, regularity and
inferential theories have been developed that offer at least tentative
solutions to the problem of spurious causation, for instance
Baumgartner (2013), Spohn (2012), and Andreas and Gunther (2020)
(which are presented in
 SS1.4,
 SS2.2,
 SS2.3,
 respectively).



Another challenge is the direction of causation. David Lewis (1973,
1979) thought that counterfactuals themselves yield an account of the
direction of causation. This account, however, turned out to be rather
controversial and is rarely adopted (see, e.g., Frisch 2014: 204n). To
determine the direction of causation, regularity and inferential
theories tend to rely on broadly Humean constraints on the temporal
order between cause and effect. This move comes at the cost of
excluding backwards-in-time causation. We must wonder how severe this
problem is. So far, there is neither a commonly agreed understanding
of the very idea of backwards causation nor actual empirical evidence
for it (cf. the entry on
 backwards causation).
 The direction of causation remains a point of controversy within
regularity, inferential, and counterfactual approaches.



To conclude, the contemporary regularity and inferential theories have
made some remarkable progress. Valid criticisms have been largely
overcome and many causal scenarios--which continue to be
challenging for counterfactual accounts of causation--have been
solved. In particular, Baumgartner (2013), Beckers and Vennekens
(2018), and Andreas and Gunther (2020) all solve the set of
scenarios including overdetermination, early and late preemption, and
switches. In light of these recent developments, it should not be
considered evident anymore that counterfactual theories of causation
have a clear edge over regularity and inferential theories.