Consider the following claims:

1. The drought caused the famine.
2. Drowsy driving causes crashes.
3. How much I water my plant influences how tall itgrows.
4. How much novocaine a patient receives affects how muchpain they will feel during dental surgery.

The metaphysics of causation asks questions about what it takes forclaims like these to be true—what kind of relation the claimsare about, and in virtue of what these relations obtain.

Although both 1 and 2 are broadly causal claims, some think that theyare not claims about the samekind of causal relation. Thesecausal relations may be differentiated by their relata. Claim 1relatestokens. It talks about aparticular droughtand famine, not droughts and faminesin general. On the otherhand, claim 2 relatestypes—it is not talking about anyparticular instance of drowsy driving, but rather drowsy drivingin general. A prominent view is that there are differentkinds of causal relation corresponding to these different kinds ofrelata. (See, for instance, Sober 1985 and Eells 1991.)

Contrast 1 and 2 with claims like 3 and 4. In claim 3, the causal verb“influences” is not flanked by token happenings, nor typesof happenings. Instead, it is flanked by what we can callvariableexpressions. Variable expressions are interrogative clauses like“how much I weigh”, “what the scale reads”,“when the game ends”, and “whether I catch thebus”. We can call the denotation of variable expressionsvariables. Just as we distinguish between token happeningsand types of happenings, we may distinguish between token variablesand type variables. For instance,how much I weigh is a tokenvariable whose value depends upon my weight.How much Barack Obamaweighs is a different token variable whose value depends uponObama’s weight. We may say that how much I exercise affects howmuch I weigh. And we may say that how much Obama exercises affects howmuch Obama weighs. These are two different claims are about causalrelations between token variables. Alternatively, we could claim thathow much one exercises affects how much one weighs. On its face, thisis not a claim about any particular variable. Instead, it is talkingabout how exercise affects weightin general. It asserts arelationship between twotypes of variables. Likewise, 4doesn’t make a claim about the relationship between anyparticular individual’s novocaine and sensation. Instead, itsays something about how novocaine affects sensationingeneral.

We will be careful to distinguish these four different kinds of causalclaims. Unfortunately, there is no standard terminology to mark thedistinction between causal claims likes like 1 & 2 and causalclaims like 3 & 4. So let us introduce new conventions. To marktheir contrast with variables, call causal relata like droughts,famines, and car crashes (whether type or token)“constants”. Then, let us call the relation which holdsbetween constantscausation. A causal claim that relatestoken constants will be called a claim abouttoken causation.(This is sometimes calledsingular causation, oractualcausation.) A causal claim that relates types of constants willbe called a claim abouttype causation. (This is sometimescalledgeneral causation.) On the other hand, the relationwhich holds between variables (whether type or token) will be calledinfluence. A causal claim that relates token variables willbe called a claim abouttoken influence. (Hitchcock 2007a,usestoken causal structure for a network of relations oftoken influence.) A causal claim that relates types of variables willbe called a claim abouttype influence.

For each of these putative causal relations, we can raise metaphysicalquestions: What are their relata? What is their arity? In virtue ofwhat do those relata stand in the relevant causal relation? And howdoes this kind of causal relation relate to the others? Of course,there is disagreement about whether each—or any—of theserelations exists. Russell (1912: 1) famously denied that there are anycausal relations at all, quipping that causation is “a relic ofa bygone age, surviving, like the monarchy, only because it iserroneously supposed to do no harm” (see also Norton 2003).Others may deny that there is a relation of general causation orinfluence at all, contending that claims like 2 and 3 are simplygeneralizations about token causal relations (see§2.1 below). There will also be disagreement about whether these relationsare reducible, and, if so, what they can be reduced to—probabilities,regularities,counterfactuals,processes,dispositions,mechanisms,agency, or what-have-you. This entry will not attempt to survey the range ofpotential answers to these questions. Instead, it will focus on moregeneral questions about the causal relata, the arity of the causalrelation, prominent or controversialinstances of thesecausal relations, how the different relations are themselves related,and so on.

• 1. Token Causation
  • 1.1 Relata
    • 1.1.1 Events
    • 1.1.2 Facts
    • 1.1.3 Variable Values
    • 1.1.4 Fineness of Grain and Causal Differences
  • 1.2 Relation
    • 1.2.1 Instances
    • 1.2.2 Arity
    • 1.2.3 Normality
• 2. Type Causation
  • 2.1 Relationship to Token Causation
  • 2.2 Net and Component Effects
• 3. Influence
  • 3.1 Relata
  • 3.2 Models
  • 3.3 Relationship to Token Causation
  • 3.4 The Metaphysics of the Models
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Token Causation

1.1 Relata

Claims like 1 describe a relation which holds betweentokencauses and effects—but what are token causes and effects? Whatare the relata of the token causal relations described by claims like1? One popular view is that token causes and effects areevents (Davidson 1963, 1967; Kim 1973; Lewis 1986b—seethe entry onevents). Others hold that token causes and effects are facts (Bennett 1988;Mellor 1995—see the entry onfacts). Within the causal modeling approach (see the entry oncausal models), it is often assumed that causes and effects are thevalues ofvariables (Hitchcock 2001a; Woodward 2003; Halpern & Pearl2005; Hall 2007; Halpern 2016a). One also finds support for otherrelata likeconditions (J. L. Mackie 1965),eventallomorphs (Dretske 1977),tropes (Campbell 1990),states of affairs (Armstrong 1997; Dowe 2000: ch. 7),situations (Menzies 1989a), andaspects (Paul 2000).Allegiances are complicated by disagreements over what events, facts,and these other creatures are. For instance, for both Bennett andMellor, facts are just states-of-affairs which obtain, bringing theirposition in line with Armstrong’s. And, amongst those who agreethat the causal relata are events, there is considerable disagreementabout what exactly events are.

1.1.1 Events

Let’s begin with events. Some have proposed that events are justregions of spacetime (Lemmon 1966; Quine 1985). That is: they proposethat events are individuated by the time and place of theiroccurrence. This is a very coarse-grained view of events. According toit, no two distinct events can occur at the same place and time. Tosee why some think this individuates eventstoo coarsely,consider the example form Davidson (1969): A metal ball is spun on ahotplate. As it rotates, the ball heats up. It heats up and rotateswithin precisely the same region of spacetime. So, if we individuateevents in terms of the time and place of their occurrence, then wemust say that the ball’s heating up and its rotation are one andthe same event. But it can seem that the ball’s heating up andits rotation differ causally. The ball’s heating up caused thesurroundings to warm, but it does not appear that the ball’srotation caused the surroundings to warm. Similarly, the ball’sbeing spun caused it to rotate, but didn’t cause it to heat up.And the hotplate caused the ball to heat up, but didn’t cause itto rotate. Moved by examples like these, Davidson (1969) suggestsindividuating events by their causes and effects. That is, Davidsonproposes thatx andy are the same event iff, forevery eventz, bothz causedx iffz causedy andx causedz iffy causedz. The left-to-right direction of thisbiconditional follows from the Indiscernability of Identicals, so theright-to-left is the substantive direction; it tells us that we shouldnot draw any more distinctions between events than are needed toaccount for differences in causation. This imposes a constraint on howa theory of events must relate to a theory of causation, but on itsown, it does not tell us what events are, nor how finely they areindividuated. After all, without some additional claims about whichevents are causally related, Davidson’s thesis is entirelyconsistent with the claim that the ball’s rotation and itsheating are identical.

Kim (1976) provides a more detailed fine-grained theory of events.According to him, events are individuated by the properties orrelations they involve, the objects which exemplify those propertiesor relations, and the times during which they exemplify thoseproperties or relations. For instance, the event of the ball rotatingis the triple of the property of rotation, the object of the ball, andthe time interval during which the ball rotates: ⟨is rotating,the ball,t₁–t₂⟩.And the event of the ball heating is the triple of the property ofheating, the object of the ball, and the time interval during whichthe ball heats: ⟨is heating, the ball,t₁–t₂⟩. These twotriples involve different properties, so they are differentevents.

Lewis (1986b) gives a different fine-grained theory of events. OnLewis’s view, events are properties of a spacetime region. ForLewis, properties are intensions, or classes of possible individuals.So, on Lewis’s view, events are classes of spacetime regions atpossible worlds. What it is for an event to occur at a world,w, is for the event to contain a spacetime region fromw. Lewis is also able to distinguish the ball’srotation from its heating; though these events occupy the same regionof spacetime at the actual world, they do not necessarily co-occur. Itis possible for the ball to heat without rotating, and it is possiblefor the ball to rotate without heating. So the class of spacetimeregions in which the ball rotates is not the same as the class ofspacetime regions in which the ball heats, and if Lewis identifies theformer class with the event of the ball rotating, and the latter withthe event of the ball heating, then he may distinguish these twoevents.

1.1.2 Facts

One reason to be unhappy with the view that token causes and effectsare events is that it appears thatabsences oromissions can be involved in causal relations. For instance,Anna’s failure to water her plant may cause it to die. Here, wehave an absence as a token cause. Likewise, Anna’s vacation mayhave caused her to not water the plant. Here, we have an absence as atoken effect. But it does not seem that absences or omissions areevents. They arenothings, non-occurrences, and are hence notidentical to any occurrent events. This motivates taking token causesand effects to befacts, rather than events. Even if there isno event which is Anna’s failing to water her plant, it isnonetheless a fact that Anna didn’t water her plant.

Some are not moved by this consideration because they deny thatabsences can be token causes and effects. For instance, Armstrongclaims

Omissions and so forth are not part of the real driving force innature. Every causal situation develops as it does as a result of thepresence of positive factors alone. (1999: 17, see also Thomson 2003,Beebee 2004a, and Moore 2009)

This position doesn’t necessarily preclude one from admittingthat absences can stand in some cause-like relation; for instance,Dowe (2000, 2001) develops an account of ersatz causation (causation*)which relates absences, even though he denies that causation properever relates absences. Others are unmoved because they think thatabsences are events. For instance, a Kimian could take Anna’sfailure to water her plant to be her exemplification of a negativeproperty (the property of not watering her plant) throughout some timeinterval. Alternatively, Hart and Honoré propose that

negative statements like “he did not pull the signal” areways of describing the world, just as affirmative statements are, butthey describe it bycontrast not bycomparison asaffirmative statements do. (1959 [1985: 38])

For example, suppose that, instead of watering the plant, Anna took astroll. Then we could take “Anna’s failure to water herplant” to be a contrastive way of describing Anna’sstroll; we could then allow that the event of Anna’s strollcaused the plant to die.

1.1.3 Variable Values

In contrast to the extensive literature on events and facts, there hasbeen comparatively less discussion of the metaphysics of variables andvariable values. When the issue is discussed, many find it congenialto reduce variable values in some way or other to one of the otherkinds of entities which have been proposed as token causal relata. Inmany applications, binary variables (variables which take on twopotential values, usually 0 and 1) are used forwhether a certainevent occurs. Then, questions about what it takes for thevariable to take on a value can be translated into questions aboutwhat it takes for the relevant event to occur. Hitchcock (2012)suggests that the values of variables be taken to be Lewsianeventalterations. (For Lewis [2000], analteration of anevent,e, is a modally fragile event—an event whichwould not occur, were it ever-so-slightly different—which is nottoo different frome itself. Some alterations ofewill be ways fore to occur, and some will be ways fore to fail to occur, but they are all alterations ofe.) An unabridged draft of Hall (2007, see reference forlink) proposes that a variable is a family of pairwise incompatiblepropositions, where each proposition is

about the state of a particular physical system or region of space ator during a particular time or time-interval.

On this understanding, a variable’s actual value justcorresponds to the true proposition in the family. If we assume thatfacts are just true propositions, we get a view on which the tokencausal relata are just facts (of a particular kind).

In general, it seems that we can take any pre-existing view abouttoken causes and effects and translate it into a view about variablevalues. For instance, take a Kimian view of events, on which, for anypropertyF, any individuala, and any time or timeintervalt, ⟨F,a,t⟩ isan event. Then, we could have a view on which each value of a givenvariable corresponds to one of these Kimian triples. What’sdistinctive about variable values as causal relata, then, isn’tthe individual values, but rather, how they are packaged together intoa single variable. For instance, taking the Kimian view as our pointof departure, we could have a variable, call itwho steals thebike, whose potential values include ⟨steals the bike,Susan,t⟩ and ⟨steals the bike, Alex,t⟩. Or we could have a variable, call itwhat Susansteals, whose potential values include ⟨steals the bike,Susan,t⟩ and ⟨steals the canoe, Susan,t⟩. And there’s a third variable, call itwhatSusan does to the bike, whose potential values include⟨steals the bike, Susan,t⟩ and ⟨buys thebike, Susan,t⟩. Now, there’s an additionalmetaphysical question faced by those who think that the token causalrelata are variable values—a question not faced byKim—which is: are the variable valueswho steals the bike= ⟨steals the bike, Susan,t⟩,what Susansteals = ⟨steals the bike, Susan,t⟩, andwhat Susan does to the bike = ⟨steals the bike, Susan,t⟩ all the same causal relatum, or are they different?(For more on these kinds of questions, see§3.1 below.)

Here is an argument (adapted from Dretske 1977) that the variablevalueswhat Susan steals = ⟨steals the bike, Susan,t⟩ andwhat Susan does to the bike =⟨steals the bike, Susan,t⟩ should be treateddifferently. Suppose that the store has water-tight security, so that,if Susan steals anything at all—be it the bike, the canoe, orwhat-have-you—she will be arrested. Then, consider the sentences5 and 6 (read them with additional stress on the emphasizedwords):

5. Susan’s stealingthe bike caused her to bearrested.
6. Susan’sstealing the bike caused her to bearrested.

As Dretske notes, while 5 sounds false, 6 sounds true. Dretske usesthis to argue for token causal relata which are more fine-grained thanevents. He calls themevent allomorphs. But, if we think thatthe causal relata are the values of variables, then it’s naturalto account for the apparent difference in truth-value between 5 and 6by contending that, while 5 is talking about a variable likewhatSusan steals, 6 is talking about a variable likewhat Susandoes to the bike. But then, in order to say that 5 is false while6 is true, we must say that the variable valuewhat Susan steals= ⟨steals the bike, Susan,t⟩ is a differentcausal relatum thanwhat Susan does to the bike =⟨steals the bike, Susan,t⟩. The former causedSusan to be arrested, whereas the latter did not.

1.1.4 Fineness of Grain and Causal Differences

We’ve now twice encountered arguments of the following form:“c causede” is true, but“c* causede” is false, so it must bethat “c” and “c*” denote twodifferent causal relata. In short: where there are differences incausation, there must be differences in the causal relata. Call thisthecausal differences argument. This argument was used toshow that token causes cannot just be regions of spacetime—forthen, the ball’s rotation and its heating would be one and thesame event, but the ball’s rotation differs causally from itsheating. It was again used to show that “Susan’s stealingthe bike” must be a different token causethan “Susan’sstealing thebike”. This second example shows us that the style of argumentcan lead to avery fine-grained view of the causal relata. Itis, after all, necessary that Susansteals thebike if and only if Susan stealsthe bike, so itlooks like this style of argument leads us to draw hyperintensionaldistinctions between causal relata. Some may see hyperintensionaldistinctions between causal relata as areductio, andconclude that something must be wrong with the causal differencesargument.

We could resist the argument in at least three different ways.Firstly, we could contend that causal claims like 5 and 6 are not infactcausal claims. Strevens (2008) proposes that apparentlycausal claims like 5 and 6 are in factcausal-explanatoryclaims. As Strevens puts it:

claims of the formc was a cause ofe…do not assert the existence of a rawmetaphysical relation between two eventsc ande;rather, they are causal-explanatory claims that assert thatcis a part of the causal explanation fore. (2008: 4)

(See also Davidson 1967, 1970; Strawson 1985.) On a view like this, wecan maintain that, whilecausation relates coarse-grainedentities like regions of spacetime,causal explanationrelates more fine-grained entities like propositions, or eventsunder-a-description.

Secondly, we could claim that “…causes…” isanintensional context, which does not allow the substitutionof co-referring terms without a change of truth-value (see Anscombe1975; Achinstein 1975, 1983; and McDermott 1995). By way ofexplanation: names of events are uncontroversially notintersubstitutable within quotation marks. From“‘Caesar’s crossing the Rubicon’ has fourwords” and “Caesar’s crossing the Rubicon = Thestart of the Roman Civil War”, we cannot conclude“‘The start of the Roman Civil War’ has fourwords”. If we think that flanking the verb “tocause” is like appearing within quotation marks in this way,then we can likewise reject the inference from “Turning on thehotplate caused the ball’s heating” and “Theball’s heating = the ball’s rotation” to“Turning on the hotplate caused the ball’srotation”.

Thirdly, we could appeal to a kind of contrastivism on which thecausal relation is four-place, rather than two-place (Schaffer 2005:§4). On this view, causal claims have the logical form“c, rather thanc*, causede, ratherthane*” (wherec* ande* arecontrast causes and effects, or perhaps sets thereof). Then, we couldsuggest that claims like 5 and 6 suggest different contrasts, whichmake a causal difference without requiring any difference in the firstor third argument places. In short: there are causal differenceswithout differences in causes or effects; some causal differences aredue to differentcontrasts. This kind of view would allow usto retain even the very coarse theory of events from Quine (1985),according to which an event is just a region of spacetime. The entrydiscusses contrastivism further in§1.2.2 below.

1.2 Relation

1.2.1 Instances

There are a wide variety of theories of the token causalrelation—theories of what it takes for one event, fact, orwhat-have-you to be a token cause of another. This entry won’tattempt to survey the available options. The interested reader shouldconsult the entries oncounterfactual theories of causation,causation and manipulability,probabilistic causation,regularity theories of causation,dispositions, andmechanisms in science. Process theories of causation are discussed in the entries oncausation in physics andWesley Salmon (see also Dowe 1997 [2008]). Instead, this entry will survey some ofthe most philosophically interesting and persistently troublesomeinstances of token causation, and discuss what theseinstances might teach us about the token causal relation.

Preemption. Cases of what’s been calledpreemption share a common structure: there is a backup,would-be cause ofe (call it “b”, forbackup). Hadc not happened, the backupbwould have been a cause ofe, butcpreemptedb, causinge to happen, and simultaneouslymaking it so thatb is not a cause ofe. Here aretwo vignettes with this structure:

1. Suzy has a grievance against her neighbor and wants to retaliateby shattering his window. Billy is also aggrieved, and tells Suzy thathe will throw the rock. So Suzy stays at home to establish an alibiwhile Billy leaves and throws a rock at the window. The rock hits andthe window shatters. Here, Suzy’s grievance is a backup,would-be cause of the window’s shattering. Had Billy not beenaggrieved, Suzy would have caused the window to shatter. But she waspreempted by Billy.
2. Patricia is suffering from a terminal disease. To ease her pain,the doctors give her a palliative dose of morphine. Due to a clericalerror, she is given too much morphine and dies from an overdose. Here,the terminal disease is a backup, would-be cause of Patricia’sdeath. Had she not been given the morphine, the disease would havekilled her. But the disease was preempted by the overdose ofmorphine.

In cases of preemption, the nearly universally shared judgment is thatthe “preempting”c is a cause ofe. Forinstance, Billy’s grievance is a cause of the window’sshattering, and the morphine is a cause of Patricia’s death.(There are dissenters to the consensus; Beckers and Vennekens (2017,2018) insist that the “preempting”c is not acause ofe in cases like these.)

The first of these vignettes is a case of what’s come to beknown asearly preemption, whereas the second is a case ofwhat’s come to be known aslate preemption. Cases ofearly preemption have roughly the same structure as the following“neuron diagram”:

Figure 1: Early preemption neurondiagram 1. [Anextended description of figure 1 is in the supplement.]

Here’s how to read this diagram: every circle represents aneuron. Associated with each neuron is a certaintime—here, the time written beneath that neuron. A neuron caneither fire or not fire at its designated time. If the circle iscolored grey, this indicates that the neuron fired. If it is coloredwhite, this indicates that the neuron did not fire. So, in the diagramabove,b,c,d, ande fired, anda did not fire. Arrows representstimulatoryconnections between neurons. If the neuron at the base of an arrowfires, then the neuron at its head will fire so long as that neuron isnot inhibited. The circle-headed lines representinhibitoryconnections between neurons. If the neuron at the base of one of theseconnections fires, then the neuron at its head willnot fire.So, in the diagram above,a doesn’t fire, even thoughb did, becausec inhibiteda.

Here, “c” stands both for the neuroncand for the event of the neuronc firing (or the fact thatc fired, or whatever), and likewise for the other letters.Then, this neuron diagram has the structure ofpreemption:b is a backup, would-be cause ofe’sfiring. Hadc not fired, the neuron system would have lookedlike this:

Figure 2: Early preemption neurondiagram 2. [Anextended description of figure 2 is in the supplement.]

Here,b is a cause ofe. So, in the original neuronsystem,b is a backup, would-be cause ofe; hadc not fired,b would have been a cause ofe, butc preemptsb and causeseitself.

As a brief aside, some authors use neuron diagrams like these asrepresentational tools for modelling the causal structure of casesdescribed by vignettes. So, they might say that the neuronbrepresents whether Suzy is aggrieved,a represents whetherSuzy throws a rock at the neighbor’s window, anderepresents whether the window shatters. Hitchcock (2007b) givesreasons to worry about this use of neuron diagrams, and argues that weshould instead use causal models as a representational tool (see theentry oncausal models, and§3.2 below). Whatever we think about using neuron diagrams to representthe causal structure of scenarios described in vignettes, there shouldbe little objection to using them to modelsystems ofneurons, hooked up with stimulatory and inhibitory connections inthe ways indicated in the diagram (Hitchcock 2007b agrees).That’s how neuron diagrams will be understood here. So theneuron diagram isn’t being used to model Suzy and Billy’ssituation. It is used to model a very simple system of five neurons,and, arguably, the system of neurons has a similar causal structure tothe vignette involving Billy and Suzy.

What makes cases of early preemptionearly is that there issome time beforee happens at which the backup causal chainis cut. In our neuron diagram, at timet2, onceafails to fire, the backupb no longer has any hope of being acause ofe’s firing. So, ifd had not fired(in violation of the neuron laws),e would not have fired. Inthe case of Billy and Suzy, once Billy tells Suzy he’s going tothrow a rock at the window, the potential causal chain leading fromSuzy’s grievance to the window’s shattering is cut. Nowthat’s she’s at home establishing an alibi, she has nohope of causing the window to shatter. Many counterfactual theories ofcausation appeal to this feature of early preemption in order tosecure the verdict thatc is a cause ofe. (See, forinstance, Ramachandran (1997, 1998), Ganeri et al. (1996, 1998), Yablo(2002, 2004), Hitchcock (2001a), Halpern and Pearl (2005), Woodward(2003), Halpern (2016a), Andreas and Günther (2020, 2021), andWeslake (ms.—see Other Internet Resources).)

Matters are different in cases oflate preemption. What makescases of late preemptionlate is that the causal chain fromthe potential backup is only cut shortafter the effecte happens. There was no time prior to her death at whichPatricia’s terminal disease stopped being deadly. So, at anytime prior to her death by overdose, were the morphine or any of itseffects to be taken away, Patricia would still have died from thedisease.

Many cases of late preemption are cases in which the causehastens the effect. That is, had the cause been absent, theeffect—or, in any case, something very similar to theeffect—would have happened later than it actually did. But thisisn’t an essential feature of the case. Consider the following:Quentin is given chemotherapy to fight his cancer. The chemotherapycompromises his immune system, and after catching a flu, Quentin diesfrom pneumonia. It is easy to suppose that the chemotherapy prolongedQuentin’s life. Suppose that, had he not received chemo, thecancer would have killed him in January. And suppose that thepneumonia killed him in February. So the chemotherapy did not hastenQuentin’s death; it delayed it. Nonetheless, this could easilybe a case of late preemption. The chemotherapy prevented the cancerfrom killing Quentin, but we may easily suppose that there is no timeprior to his death at which removing the chemotherapy, or the flu, orthe pneumonia, would have prevented him from dying. We may easilysuppose that the cancer remained deadly throughout. (For more oncauses, hasteners, and delayers, see Bennett 1987; Lombard 1990; P.Mackie 1992; Paul 1998a; Sartorio 2006; Hitchcock 2012; and Touborg2018.) The case of Quentin can easily be modified so that, had he notreceived chemo, he would have died at exactly the same time that heactually died. This version gives us a case of late preemption inwhich, had the cause been absent, the effect—or, in any case,something very similar to the effect—would have happened at thevery same time that it actually did.

Cases of preemption show us that causes need not benecessaryfor their effects. The effecte did not depend upon its causec. Hadc been absent,e would still havehappened, due to the backupb. They moreover suggest thatthere is something important about the process whereby causes andeffects are connected. Bothc and the backupb weresufficient fore to happen. What seems to make the differencebetween them is that there is the right kind of connecting processleading fromc toe, and there is not the right kindof connecting process leading fromb toe. For thisreason, while counterfactual, probabilistic, agential, and regularitytheories often stumble on cases of preemption, they are more easilytreated by process theories.

Schaffer (2000a) introduces cases of what he callstrumpingpreemption. In these cases, there is preemption even though thereis no “cutting” of the backup causal process. That is:there is no missing part of the process leading from the backupb toe, but nonetheless,b does not countas a cause ofe becausec “trumps”b. Here are two cases like this:

1. The laws of magic say that the first spell cast on a given daygoes into effect at midnight; spells cast after the first areineffective. On Monday, there are only two spells cast. In themorning, Merlin casts a spell to turn the prince into a frog. And inthe evening, Morgana casts a spell to turn the prince into a frog. Atmidnight, the prince turns into a frog.
2. The Major outranks the Sergeant, who outranks the Corporal. TheCorporal will follow the orders of the highest-ranking officer. TheMajor and the Sergeant both give the Corporal the order to advance,and the Corporal advances.

In the first case, Schaffer contends that Merlin’s spell causedthe prince to turn into a frog and that Morgana’s spell did not.Merlin’s spell, after all, was the first spell of the day, andthat’s what the laws of magic identify as the relevant feature.And, in the second case, Schaffer contends that the Major caused theCorporal to advance, and the Sergeant did not. After all, the Corporallistens to the orders of the highest ranking officer, and in thiscase, that’s the Major. (For further discussion, see Lewis 2000;Halpern & Pearl 2005; Hitchcock 2007a; Halpern & Hitchcock2010: §4.2; and Paul & Hall 2013: §4.3.4.)

Prevention.Cases of preemption tend to be handledeasily by process theories of causation, whereas counterfactual,manipulation, probabilistic, and regularity theories have moredifficulty saying both that the preemptingc is a cause ofe and that the preempted backupb is not a cause ofe. For, in cases of preemption, we can easily trace out theprocess wherebyc leads toe, whereas the processwhich would have led fromb toe was interrupted byc. On the other hand, process theories have more difficultywith cases ofprevention. Here’s a case of prevention:over the Atlantic ocean, James Bond shoots down a missile which isheading for Blofield’s compound, and Blofield survivesunscathed. Bond prevented Blofield from dying. If we understandprevention as causation-not (c preventse iffc caused it to be the case thate didn’thappen), then Bond caused Blofield to not die. But there does notappear to be any causalprocess leading from Bond toBlofield—both Bond and the missile were thousands of milesaway.

Some deny that cases of prevention are genuinely causal (see Aronson1971; Dowe 2000; Beebee 2004a; and Hall 2004). This response could befurther justified with worries about absences being effects (recall§1.1.2). However, there are additional worries about the cases which Hall(2004) calls “double prevention”. In these cases,c preventsd, and, ifd had happened, thend would have preventede. Soc prevents apreventer ofe. For instance: Blofield launches a cyberattack on Europe’s electrical grid, plunging the continent intothe dark. Had Bond not shot down the missile, it would have preventedBlofield from carrying out the cyber attack. So Bond prevented apotential preventer of the cyber attack. Here, there is an inclinationto say that Bond was an (inadvertent) cause of the cyber attack.However, there is no connecting process leading from Bond to the cyberattack—there is no relevant energy-momentum flow, marktransmission, or persisting trope connecting them.

Cases of double prevention have roughly the structure of this neurondiagram:

Figure 3: Double prevention neurondiagram. [Anextended description of figure 3 is in the supplement.]

As Schaffer (2000c, 2012b) argues, many paradigm cases of causationturn out, on closer inspection, to be instances of double prevention.Pam uses a catapult to hurl a rock through the window. It seems clearthat Pam’s actions caused the window to shatter. But suppose thecatapult works like this: a catch prevents the catapult fromlaunching, and Pam’s releasing the catch prevents it frompreventing the launch. Thus, the relationship between Pam’sreleasing the catch and the shattering of the window is one of doubleprevention. Here, it is less comfortable to deny that Pam caused thewindow to shatter, though not all agree. Aronson denies that there isany causation in a similar case:

Consider a weight that is attached to a stretched spring. At a certaintime, the catch that holds the spring taut is released, and the weightimmediately begins to accelerate. One might be tempted to say that therelease of the catch was the cause of the weight’s acceleration.If so, then what did the release of the catch transfer to the weight?Nothing, of course. (1971: 425)

Aronson contends that, while the release of the catch was an enablingcondition for the weight’s acceleration, it was not strictlyspeaking acause of the weight’s acceleration.

Preemptive Prevention. There’s anotherinteresting class of cases where one preventerpreemptsanother (see McDermott 1995 and Collins 2000). Here are two cases likethis:

1. Either Bond or M could shoot down the missile headed forBlofield’s compound. Bond tells M that he will do it, so M goeshome. Bond shoots down the missile, and Blofield survives.
2. Bond shoots down a missile headed for Blofield’s compound.However, Blofield’s compound has its own perfectly reliableanti-missile defense system. So, had Bond not shot the missile down,the anti-missile defense system would have, and Blofield would havesurvived unscathed.

By analogy with ordinary preemption, we can call the first case aninstance ofearly preemptive prevention. Once Bond tells Mthat he will shoot down the missile, M is no longer a potentialpreventer of the missile strike. If, at the last minute, Bond hadchanged his mind, Blofield’s compound would have been destroyed.In contrast, we can call the second case an instance oflatepreemptive prevention. At no point does the missile defensesystem “step down”, and stop being a potential preventerof the compound’s destruction.

The first case has a structure similar to this neuron system,

Figure 4: Early preemptive preventionneuron diagram. [Anextended description of figure 4 is in the supplement.]

Whereas the second case has a structure similar to this neuronsystem,

Figure 5: Late preemptive preventionneuron diagram. [Anextended description of figure 5 is in the supplement.]

There seems to be more of an inclination to attribute causation incases of early preemptive prevention than in cases of late preemptiveprevention. As McDermott (1995) puts it: in case (2), many areinitially inclined to deny that Bond prevented the compound’sdestruction; but, when they are asked “Of Bond and the missiledefense system, which prevented the compound’sdestruction?”, it feels very natural to answer with“Bond”. (As Collins 2000 notes, things begin to feeldifferent if we suppose that the anti-missile defense system is lessthan perfectly reliable.)

Switches. Suppose that a lamp has two bulbs: one onthe left, and one on the right. There is a switch which determineswhether power will flow to the bulb on the left or the one on theright. If the power is on and the switch is set to the left, then theleft bulb will be on, and the room will be illuminated. If the poweris on and the switch is set to the right, then the right bulb will beon, and the room will be illuminated. If the power is off, thenneither bulb will be on, and the room will be dark, no matter how theswitch is set. To start, the power is off and the switch is set to theleft. Filipa flips the switch to the right, and Phoebe turns on thepower. The right bulb turns on, and the room is illuminated.

Cases like these are discussed by Hall (2000) and Sartorio (2005,2013), among others. This particular example comes from Pearl (2000).In these kinds of cases, there are five events (or facts, orwhat-have-you):f,p,l,r, ande, with the following features: ifp happens, itwill cause eitherl orr to happen, depending uponwhetherf happens. And if eitherl orrhappens, it will causee to happen. For instance, in ourexample,f is Filipa flipping the switch to the right,p is Phoebe turning on the power,l andrare the left and right bulbs turning on, respectively, andeis the room being illuminated. In this case, it can seem thatthere’s an important difference between Filipa and Phoebe.Whereas Filipa made a difference tohow the room gotilluminated (whether by the left bulb or the right one), she did notmake a difference towhether the room got illuminated.Phoebe, in contrast, did make a difference towhether theroom got illuminated. It seems natural to say that, whilePhoebe’s turning on the power was a cause of the room beingilluminated, Filipa’s flipping the switch was not.

Switching cases have roughly the same structure as the followingneuron diagram:

Figure 6: Switch neuron diagram 1. [Anextended description of figure 6 is in the supplement.]

Here, the neurons (the switch) is special. It can either beset to the left or to the right, indicated by the direction the arrowis pointing. Likewise, the connection betweenf ands is special. Iff fires, thens will beset to the right. Iff does not fire, thens will beset to the left. On the other hand, the connection betweenpands is normal. Ifp fires, thens willfire. Ifs fires while set to the left, thenl willfire. Ifs fires while set to the right, thenr willfire. If eitherl orr fires, thene willfire.

Iff hadn’t fired,s would have fired whileset to the left, sol would have fired, ande wouldhave fired.

Figure 7: Switch neuron diagram 2. [Anextended description of figure 7 is in the supplement.]

On the other hand, ifp hadn’t fired,s wouldstill have been set to the right, but it would not have fired, soneitherr nore would have fired:

Figure 8: Switch neuron diagram 3. [Anextended description of figure 8 is in the supplement.]

Reflection on switches can lead to the conclusion that token causationcannot be merely a matter of the intrinsic nature of the processleading from cause to effect. Consider the following variant: whilethe right bulb is functional, the left bulb is not. If the power hadbeen turned on while the switch was set to the left, the left bulbwould not have turned on, and the room would have remained dark. Giventhis setup, Filipa’s flipping the switch to the rightdoes appear to be a cause of the room’s beingilluminated. After all, with this setup, had Filipa not flipped theswitch, the room would have remained dark. But, if we just look at theintrinsic features of the process leading from Filipa’s flippingthe switch to the room’s illumination, there will be nodifference. Or consider the following system of neurons:

Figure 9: Switch neuron diagram 4. [Anextended description of figure 9 is in the supplement.]

Here,f’s firing appears to be a cause ofe’s firing (along withp). Iffhadn’t fired, then the switchs would have been set tothe left, soe would not have fired. Sof’sfiring was needed fore to fire. However, there doesn’tseem to be any difference between the process leading fromf’s firing toe’s firing inthis system of neurons and the process leading fromf’s firing toe’s firing in theoriginal system of neurons.

So switches suggest that whether one event (fact, or whatever),c, is a cause of another,e, is not just a matter ofthe intrinsic features of the process leading fromc toe. It looks like we may also have to consider counterfactualinformation about what things would have been like in the absence ofc. (See the discussion in Paul and Hall, 2013.)

Switches also pose a problem for the view that causation istransitive—that is, the view that, ifc causesd andd causese, thenc causese. For, in the original version of the case, it seems thatFilipa’s flipping the switch to the right caused the right bulbto turn on. And it appears that the right bulb turning on caused theroom to be illuminated. But it does not appear that Filipa’sflipping the switch to the right caused the room to be illuminated.(For further discussion of switches and the transitivity of causation,see McDermott 1995; Hall 2000, 2007; Paul 2000; Hitchcock 2001a; Lewis2000; Maslen 2004; Schaffer 2005; Halpern 2016b; Beckers &Vennekens 2017; and McDonnell 2018.)

1.2.2 Arity

The standard view is that causation is a binary relation between tworelata: cause and effect. However, some suggest that causation may bea ternary or a quaternary relation. Inspired by van Fraassen(1980)’s work on contrastive explanation, we could takecausation to be a ternary, or 3-place, relation between a cause, aneffect, and a contrast for the effect. On this view, the logical formof the causal relation is:c is a cause ofe, ratherthane*. For instance, it seems correct to say thatAdam’s being hungry caused him to eat the apple, rather thantoss it aside. But it seems wrong to say that Adam’s beinghungry caused him to eat the apple, rather than the pear. So changingthe contrast for the effect seems to makes a difference to causation.Hitchcock (1993, 1995a, 1996) gives a contrastive probabilistic theoryof causation, according to which causation is a ternary relationbetween a cause, a contrast for the cause, and an effect. On thisview, the logical form of the causal relation is:c, ratherthanc*, is a cause ofe. For instance, it can seemthat Paul’s smoking a pack a day, rather than not at all, is acause of his lung cancer. But it seems wrong that Paul’s smokinga pack a day, rather than two packs a day, is a cause of his lungcancer. Schaffer (2005) combines these two views, arguing thatcausation is a quaternary relation between a cause, a contrast for thecause, an effect, and a contrast for the effect. On this view, thelogical form of the causal relation is:c, rather thanc*, is a cause ofe, rather thane*.

If we think that the causal relation is ternary or quaternary, thensome of the arguments we considered earlier can look weaker. Forinstance, consider what we called thecausal differencesargument from§1.1.4. We argued that we must distinguish the cause “Susan’sstealingthe bike” from “Susan’sstealing the bike”, since the latter, but not theformer, caused her to be arrested. If we are contrastivists aboutcausation, then we could insist that the differences in focal stressserve to make salient different contrasts for one and the same event:Susan’s stealing of the bike. When we emphasize “thebike”, we suggests a contrast event in which Susan stealssomething else. And Susan’s stealing the bike, rather than thecanoe, didn’t cause her to be arrested. On the other hand, whenwe emphasize “stealing”, we suggest a contrast event inwhich Susan does something else to the bike—purchasing it,let’s say. And Susan’s stealing the bike, rather thanpurchasing it,is a cause of her arrest. Schaffer (2005) evensuggests that, if causation is 4-place, then we could take theincredibly coarse-grained Quinean view of events.

Schaffer (2005) additionally suggests that, if we say that causationis quaternary, we can defend a version of transitivity. Of course,transitivity is a property of a binary relation, but Schaffer proposesthat the quaternary relation satisfies the following constraint: ifc, rather thanc*, is a cause ofd, ratherthand*, andd, rather thand* is a causeofe, rather thane*, thenc, rather thanc* is a cause ofe, rather thane*. Thinkof it like this: corresponding to the 4-place causal relation is atwo-place relation between event-contrastpairs; and Schaffermaintains that this two-place relation between pairs of events andtheir contrasts is a transitive relation. It’s not immediatelyobvious how this helps with switches like the one we saw in§1.2.1 above. For it seems that Filipa’s flipping the switch to theright, rather than not flipping it, caused the right bulb to be on,rather than off. And it seems that the right bulb being on, ratherthan off, caused the room to be illuminated, rather than dark. But itdoesn’t appear that Filipa’s flipping the switch to theright, rather than not flipping it, caused the room to be illuminated,rather than dark. The trick is that Schaffer has a view of events onwhich they areworld-bound. So, in order to make the firstcausal claim true, the contrast “the right bulb’s beingoff” must be an event which only occurs in the world in whichFilipa does not flip the switch to the right, and the left bulb is on.Therefore, the second causal claim will turn out to be false; theright bulb’s being on, rather than offin a world in whichthe left bulb is on, is not a cause of the room’s beingilluminated, rather than dark.

1.2.3 Normality

It is natural to distinguish betweencauses andbackground, orenabling, conditions. For instance,suppose that you strike a match and it lights. It’s natural tocite your striking the match as a cause of its lighting, butit’s far less natural to cite the presence of oxygen as a causeof its lighting, even though, without the oxygen, the matchwouldn’t have lit. There’s some inclination to say thatthe presence of oxygen merelyenabled the strike to cause thematch to light, but did not cause it to light itself. Lewis (1973)echoes a traditional, dismissive, attitude when he refers to thedistinction between causes and background conditions as“invidious discrimination” (see also Mill 1843 and J. L.Mackie 1974). On this view, the distinction is pragmatic,unsystematic, and dependent upon our interests. Alien scientists fromVenus—where there is no oxygen in the atmosphere—mightfind it incredibly natural to point to the presence of oxygen as acause of the fire. On the traditional view, neither we nor theVenusian scientists are making any objective error; we each simplyhave different expectations about the world, and so we find somefeatures of it more noteworthy than others. There are ever-so-manycauses out there, but weselect some of them to call causes.The others, the ones we do not select, are regarded as mere backgroundconditions.

Hart and Honoré (1959 [1985]) suggest that conditions which arenormal, ordefault, are background conditions;whereas those which areabnormal, ordeviant, arecauses. This distinction between normal, default, or inertialconditions and abnormal, deviant, or non-inertial causes has beenappearing with increasing regularity in the recent literature. Forinstance, McGrath (2005) presents the following vignette: Abigail goeson vacation and asks her neighbor, Byron, to water her plant. Byronpromises to water the plant, but doesn’t, and the plant dies. Itsounds very natural to say that Byron’s failure to water theplant was a cause of its death; it sounds much worse to suggest thatAbigail’sother neighbor, Chris, caused the plant todie by not watering it. However, it seems that the only relevantdifference between Bryon and Chris is that Byron made apromise to water the plant, and Chris did not. SoByron’s failure to water the plant was abnormal, whereasChris’s failure to water the plant was normal. McGrath’sexample involves absences or omissions, but this feature of the caseis incidental. Hitchcock and Knobe (2009) give the following case:while administrators are allowed to take pens, professors are not.Both Adriel the administrator and Piper the professor take pens. Laterin the day, there are no pens left, and the Dean is unable to sign animportant paper. Here, it seems much better to cite Piper than Adrielas a cause of the problem. And it seems that the only relevantdifference is that Adriel waspermitted to take the pen, butPiper was not. (For more, see Kahneman & Miller 1986; Thomson2003; and Maudlin 2004.)

Considering just these kinds of cases, it’s not unnatural to seethe phenomenon as just more “selection effects”. Perhapsboth Byron and Chris caused the plant to die, but for pragmaticreasons we find it more natural to cite the person who broke a promiseas a cause. Hall (2007) presents a more challenging argument for theview that normality and defaults should figure in our theory ofcausation. He asks us to consider the following pair of neuronsystems:

(a)

(b)

Figure 10: Pair of neuron diagrams. [Anextended description of figure 10 is in the supplement.]

In figure 10(a), we have the case ofearly preemption from§1.2.1. Here,c’s firing is a cause ofe’sfiring. In figure 10(b), we have a case of what Hall calls a“short circuit”. Whenc fires, it both initiatesa threat toe’s dormancy (by makinga fire)and neutralizes that very threat (by makingd fire).Here’s a vignette with a similar structure (see Hall 2004, andHitchcock 2001a): a boulder becomes dislodged and careens down thehill towards Hiker. Seeing the boulder coming, Hiker jumps out of theway, narrowly averting death. In this case, the boulder’s fallboth initiates a threat to Hiker’s life and neutralizes thatvery threat (by alerting them to its presence). Hall contends that theboulder’s becoming dislodged did not cause Hiker to survive;and, in the case of the short circuit,c’s firing didnot causee to not fire. Intuitively,c accomplishednothingvis-a-vise. But Hall notes that these twoneuron systems are isomorphic to each other. We can make the pointwith systems of structural equations (see the entry oncausal models, and§3.2 below). Start with the case of early preemption, and use binaryvariables,A,B,C,D, andE for whethera,b,c,d,ande fire, respectively. These variables takes on the value1 if their paired neuron fires, and they take on the value 0 if theydo not. Then, we can represent the causal structure of earlypreemption with this system of equations:

Turning to the short circuit, notice that we can useA*,B*, andE* as binary variables for whether theneuronsa,b, andedo not fire.These variables take on the values 1 if their paired neurondoesn’t fire, and they take on the value 0 if itdoes. And, again, we can useC andD asvariables for whether the neuronsc andd fire;these non-asterisked variables take on the value 1 if their pairedneurondoes fire, and 0 if itdoesn’t. Withthese conventions in place, we can then write down the followingisomorphic system of equations for the neuron system of shortcircuit:

These equations are isomorphic to the ones we wrote down for the caseof early preemption. Moreover, the values of the variables areprecisely the same in each case. In the case of early preemption,\(A=0\) and \(B = C = D = E = 1\); whereas, in the case of shortcircuit, \(A^*=0\) and \(B^* = C = D = E^* = 1\) (similar cases arediscussed in Hiddleston 2005).

Therefore, if we want to claim thatc is a cause ofe in the case of early preemption, but deny thatcis a cause ofe in the case of short circuit, we will have topoint to some information which is not contained in these equationsand the values of these variables. And several authors have reachedfor the distinction betweendefault,normal statesanddeviant, abnormal events (see, for instance, Hall 2007;Hitchcock 2007a; Halpern 2008, 2016a; Paul & Hall 2013; Halpern& Hitchcock 2015; and Gallow 2021. For criticism, see Blanchard& Schaffer, 2017).

On a traditional picture, there is a broad, indiscriminate notion ofcausation which makes no appeal to normality; considerations ofnormality are used, pragmatically, to “select” which ofthe indiscriminate causes we call “causes” and which wecall “background conditions”. The isomorphism betweenearly preemption and short circuit challenges this picture, for tworeasons. Firstly,c’s firing is presumably just asdeviant in early preemption as it is in short circuit, so itcan’t just be the normality of the putative cause which makesthe difference. Secondly, those who want to deny thatc’s firing is a cause ofe’s failure tofire in short circuit do not want to claim thatc’sfiring was even a background condition fore’s failureto fire. The inclination is to say thatc’s firing hadnothing at all to do withe’s failure tofire.

2. Type Causation

2.1 Relationship to Token Causation

Type causal relations are described by generic claims like“Drowsy driving causes crashes” or “Smoking causescancer”. One prominent question concerns the relationshipbetween type and token causal claims. One view has it that type causalclaims are just generalizations or generics about token causal claims(see Lewis 1973; Hausman 1998: chapter 5; and Hausman 2005). Forinstance, to say that smoking causes cancer is just to say that tokencancers are generally caused by token histories of smoking. And to saythat drowsy driving causes crashes is just to say that, in general,token episodes of drowsy driving cause token crashes. Generic claimslike these shouldn’t be understood as saying thatmostor evenmany token episodes of drowsy driving are tokencauses of car crashes. Compare: “mosquitos carry the West Nilevirus”, which is a true generic despite the fact that mostmosquitos do not carry the West Nile virus. (See the entry ongenerics.) On this view, type causal claims are ultimately about token causalrelations.

Another view has it that type causal relations are more fundamentalthan token causal relations; what makes it the case thatChris’s smoking caused his cancer is, at least in part,that smoking causes cancer, Chris smoked, and he got cancer. This kindof view is defended by theorists like Hume (1739–40, 1748), Mill(1843), J. L. Mackie (1965), Hempel (1965), and Davidson (1967). Thesetheorists begin by giving an analysis of causal relations betweentypes of events, facts, or what-have-you. For instance: Hume says thatwhat it is for the typeC to cause the typeE is forthings of typeC to be constantly conjoined with things oftypeE. This general regularity is then used to explain whyit is that any particular thing of typeC is a token cause ofany particular thing of typeE. Subsequent regularity and“covering law” theories add additional bells and whistles;but they retain the idea that a token causal relation betweenc ande holds in virtue of some broader regularityor law which subsumes the particular relationship betweencande.

One reason to doubt that token causal relations are justinstantiations of type causal relations is that there appear to betoken causal relations without any corresponding type causal relation,and there appear to be token causal relations which go against thecorresponding type causal relations. Scriven (1962) gives thefollowing example: you reach for your cigarettes, accidentallyknocking over an ink bottle and staining the carpet. Your reaching foryour cigarettes caused the carpet to be stained, but it’s notthe case that reaching for cigarettes causes carpet stains in general.Suppes (1970) attributes this example to Deborah Rosen: at the golfcourse, you hit the ball into a tree, the ball rebounds and,fantastically enough, goes into the hole. In this case, hitting theball into the tree caused the hole-in-one, but hitting the ball intotrees does not cause holes-in-one in general.

A third position, defended by Ellery Eells (1991), is that neithertoken nor type causal relations are more fundamental than the other.Eells gives two probabilistic analyses of causation: one for typecausation, and another for token causation. Eells thinks that typecausal claims cannot just be generalizations about token causal claimsbecause of examples like this: drinking a quart of plutonium causesdeath, even though nobody has ever drank a quart of plutonium, and sono particular persons’s death has been caused by drinking aquart of plutonium. So this type causal claim cannot be ageneralization over token causal claims. (See Hausman 1998: chapter 5,for a response.)

For more on the relationship between token and type causation,particularly within a probabilistic approach to causation, seeHitchcock (1995a).

2.2 Net and Component Effects

Consider the following case, adopted from Hesslow (1976): suppose thatbirth control pills (B) prevent pregnancy (P).However, they can also have the unintended side-effect of thrombosis(T). So we might be inclined to accept the type causal claim“birth control pills cause thrombosis”. But wait:another potential cause of thrombosis is pregnancy. And birthcontrol pills inhibit pregnancy. The general structure of the case isas shown below.

Figure 11: Hesslow’s birth controlcase [Anextended description of figure 11 is in the supplement.]

Birth control pillsdirectly promote thrombosis, but at thesame time, they prevent pregnancy, which itself promotes thrombosis.By adjusting the probabilities, we can make it so that, overall,taking birth control makes no difference to the probability ofthrombosis. Then, we might be inclined to accept the type causal claim“birth control pills have no effect onthrombosis”—after all, your chances of getting thrombosisare exactly the same, whether or not you take birth control. Hitchcock(2001b) argues that cases like this require us to distinguish twodifferent kinds of effects which type causes can have: what he calls anet effect, on the one hand, and what he callscomponenteffect, on the other. In Hesslow’s case, birth controlpills have nonet effect on thrombosis. This is the sense inwhich it’s true to say “birth control pills have no effecton thrombosis”. At the same time, birth control pills have acomponent—orpath-specific—effect onthrombosis. Along the pathB →T, birth controlpromotes thrombosis. This is the sense in which it’s true to say“birth control pills cause thrombosis”. For furtherdiscussion, see Woodward (2003: §2.3) and Weinberger (2019).

3. Influence

3.1 Relata

As emphasized in§1.1 above, any theory of events or facts can easily be transposed into atheory of the values of token variables. And, plausibly, tokenvariables are individuated by their values. So, when it comes to tokeninfluence between variables, we will face many of the same questionsand debates about the causal relata: how fine- or coarse-grained arethey? When are two variables the same; when are they different? Canvariables include absences or omissions as values? There is, however,an additional question to be a asked about the metaphysics ofvariables: when can some collection of variablevalues begrouped together into a single variable? All agree that the potentialvalues of a variable mustexclude each other. That is: if\(v\) and \(v^*\) are two values of the variable \(V\), then it shouldbe impossible that \(V=v \wedge V=v^*\) (here, \(V=v\) is theproposition that the variable \(V\) takes on the value \(v\), andlikewise \(V=v^*\) is the proposition that the variable \(V\) takes onthe value \(v^*\)).

But there are additional questions to be asked about which values maybe grouped together into token variables. For instance: must thevalues of a variable concern the same time? Or can a single variablehave multiple values which concern different times? Suppose Patriciadied on Monday from an overdose of morphine; had she not received themorphine, she would have died on Tuesday. Then, there are severaldifferent variables we might try to use when thinking aboutPatricia’s death. We could use a single variable forwhetherPatricia died. This variable would take on one value if Patriciadied on Monday or Tuesday, and another value if she remained alivethroughout Monday and Tuesday. Or we could use a single variable forwhen Patricia died. This variable would take on one value ifshe died on Monday, and another value if she died on Tuesday.Alternatively, we could use a collection of time-indexed variables,whether Patricia had died by timet, for some rangeof timest. As Hitchcock (2012) puts it: the question iswhether Patricia’s dying on Monday and Patricia’s dying onTuesday

correspond to thesame value of thesame variable,todifferent values of thesame variable, or tovalues of differentvariables. (2012: 90)

Of course, this question carries the presupposition that we mustchoose between these options. You might instead think thatall of these variables exist, and each enter into differentrelations of influence.

When it comes to type variables, there are additional questions abouthow token variables are to be type-individuated, over and abovequestions about the individuation conditions of the token variablesthemselves. This question is not much discussed, but it is natural tothink that the type individuation of variables is inherited from thetype-individuation of their values. That is: \(X_1\) and \(X_2\) areof the same type if and only if \(X_1\) and \(X_2\) have the samenumber of potential values, and each those values are of the sametype. For instance, the token variableshow much I weigh andhow much Obama weighs are of the same type, as they both havethe same type of potential values (1 kg, 2 kg, etc); on the otherhand,how much I weigh andhow many pounds of arugulaObama eats are not of the same type, since their potential valuesare not of the same type.

3.2 Models

Relations of influence between variables are often encoded in formalmodels. Just as the causal relations between tokens and types can beeither deterministic or probabilistic, so too can the relations ofinfluence between variables be either deterministic or probabilistic.Deterministic relations between variables are represented withstructural equations models; whereas indeterministic relations betweenvariables can be represented with probabilistic models (sometimes withstructural equations models with random errors—see the entry oncausal models—or often with just a causal graph paired with a probability distributionover the values of the variables appearing in that graph—see theentry onprobabilistic causation.)

In the deterministic case, the relations of influence betweenvariables can be encoded in a system of structural equations. Forillustration, consider again the neuron system we used in§1.2.1 above to illustrate early preemption. As we saw in§1.2.3, for each neuron in the system, we can introduce a variable forwhether that neuron fires at the relevant time. Once we’ve doneso, the following system of structural equations describes the causalrelations between these variables.

Before getting to the metaphysical questions about what kinds ofrelations these equations represent, and what it takes for a system ofequations like this to be correct, let us first focus on how theseequations areused in practice. It’s important torecognize that there is meant to be a difference betweenstructural equations and ordinary equations. The equation \(D= C\) is symmetric. It could be re-written, equivalently, as \(C=D\).In contrast, thestructural equation \(D \coloneqq C\) isnonsymmetric. It tells us that the variable \(D\) is causallyinfluenced by the variable \(C\). And this form of causal influence isnot symmetric. To emphasize that these equations are not symmetric,“\(\coloneqq\)” is used, rather than“\(=\)”.

Some terminology: given a system of structural equations, thevariables which appear on the left-hand-side of some equation arecalled theendogenous variables. The variables which onlyever appear on the right-hand-side of the equations are calledexogenous. The model does not tell us anything about how thevalues of the exogenous variables are determined; but it does tell ussomething about how the values of the endogenous variables arecausally determined by the values of the other variables in themodel.

In this case, an assignment of values to the exogenous variables isenough to tell us the value of every endogenous variable in the model.That is: if you know that \(B=C=1\), you also know that \(A=0\) and\(D=E=1\). That needn’t be the case in general. For instance,consider the following system of equations:

In this system of structural equations, even after you know that theexogenous variable \(Z = 10\), you are not able to solve for thevalues of the two endogenous variables, \(X\) and \(Y\). The reasonfor this is that, in this system of equations, there is acycle of influence: \(X\) influences \(Y\), and \(Y\)influences \(X\). When there are cycles of influence like this, even adeterministic system of equations, together with an assignment ofvalues to the exogenous variables, will not determine the values ofall of the endogenous variables. (See the entry onbackwards causation and the section on causal loops in the entry ontime travel.) However, if we rule out cycles of influence like this, then anassignment of values to the exogenous variables will determine thevalues of all of the variables in the model. Likewise, any probabilitydistribution over the exogenous variables will induce a probabilitydistribution over the endogenous variables, as well.

It is often assumed that each of the structural equations in thesystem is independently disruptable. For instance: there is, at leastin principle, some way of disrupting \(C\)’s causal influence on\(D\)—some way of making it so that the structural equation \(D\coloneqq C\) no longer holds—which leaves it so that both ofthe structural equations \(A\coloneqq B \wedge \neg C\) and\(E\coloneqq A \vee D\) continue to hold. Not every way of disruptingthe causal relation between the variables \(C\) and \(D\) will be likethis. For instance, suppose that we remove all of the connectionsemanatingout of the neuronc. This will disrupt thecausal connection between \(C\) and \(D\), but it will also disruptthe causal connection between \(C\) and \(A\); it will make it so thatthe structural equation \(A\coloneqq B \wedge \neg C\) no longerholds. (This equation tells us that, if \(C=1\), then \(A=0\); but,with the connection betweenc anda severed, this isno longer true.) This property of a system of structuralequations—that each of the equations may be disrupted withoutaffecting any of the other equations in the model—is known asmodularity. (For criticism of this requirement, seeCartwright 2002; for a defense, see Hausman & Woodward 1999, 2004.For more on modularity, see Woodward 2003.)

If the system of equations is modular, then it is at least inprinciple possible to disrupt one of the equations without affectingany of the others. Suppose that happens, and we disrupt the equation\(A\coloneqq B \wedge \neg C\) without affecting any of the otherequations, for instance. Suppose further that we do this in such a wayas to determine the value of \(A\), or to determine a probabilitydistribution over the values of \(A\). Then, we have performedanintervention on the variable \(A\). Note that this notion of anintervention is relative to a system of equations. Whether some way ofdisrupting an equation and directly setting the value of the variableon its left-hand-side counts as an intervention or not will vary fromcausal model to causal model. In general,an intervention onan endogenous variable, \(V\) (relative to some model) is some way ofmaking it so that \(V\)’s structural equation (the equation with\(V\) on the left-hand-side) no longer holds, even though every otherstructural equation in the model continues to hold, and directlysetting the value of \(V\), or directly determining a probabilitydistribution over the values of \(V\).

Given a deterministic causal model—a system of structuralequations—the way to formally represent an intervention on anendogenous variable, \(V\), is straightforward: you remove thestructural equation which has that variable on its left-hand-side, andleave all the other equations unchanged. You go on to treat \(V\) asif it were an exogenous variable, with the value or the probabilitydistribution it was given through the intervention. Assuming thesystem of equations is acyclic, you can then work out the value of theother variables in the model, or the probability distribution over thevalues of the other variables in the model, as before.

Interventions like these have been used by many to provide a semanticsfor what will here be calledcausal counterfactualconditionals. As the term is used here, what makes a counterfactualcausal is that it holds fixed factors which are causallyindependent of its antecedent. It doesn’t say anything aboutwhat would have tohave been different in order for theantecedent to obtain. That is: it says nothing about the necessarycausal precursors of the antecedent. It holds fixed all factors whichare not causally downstream of the antecedent, and only allows toswing free factors which are causally downstream of the antecedent.Within a system of structural equations, this is accomplished bymodeling anintervention to bring about the truth of theantecedent. (For more on this “interventionist” treatmentof counterfactuals, see Galles & Pearl 1998; Briggs 2012; Huber2013, and the entry oncounterfactuals.)

3.3 Relationship to Token Causation

Several authors have provided theories of token causation which usecausal models like these. (See, for instance, Hitchcock 2001a, 2007a;Woodward, 2003, Menzies 2004, 2006 [Other Internet Resources]; Halpern & Pearl 2005; Hall 2007; Halpern 2008, 2016a, 2016b;Beckers & Vennekens 2017, 2018; Andreas & Günther 2020,2021; Gallow 2021; Weslake ms.—see Other Internet Resources.)The theories almost always understand the causal models as describingrelations of influence betweentoken variables. Most of thesetheories are roughly counterfactual. They attempt to use theinterventionist semantics for causal counterfactuals to provide anaccount of when one variable value is a token cause of another. Onthis approach, the networks of influence encoded in a causal modelprovide the pathways along which token causation propagates. If onevariable value, \(C=c\), is going to be a token cause of another,\(E=e\), then there must be some path of influence leading from thevariableC to the variableE,

The theories diverge in what additional conditions must be met for\(C=c\) to be a token cause of \(E=e\). Many, though not all, agreethat counterfactual dependence between \(C=c\) and \(E=e\) issufficient for \(C=c\) being a token cause of \(E=e\). (For more, seethe entry oncounterfactual theories of causation.)

3.4 The Metaphysics of the Models

This section discusses what it takes for these models to beaccurate—what the world must be like to host the relations ofinfluence described by the model. However, it must be noted that notevery author in the literature would be happy with the way thisquestion is posed. Many prefer to talk about whether a model isappropriate orapt. This is in part because theyrecognize that many of the models written down and used in practicemisrepresent the world. For instance, the models portray systems asdeterministic, neglecting minute quantum mechanical chances. Thus,Halpern writes that

I am not sure that there are any “right” models, but somemodels may be more useful, or better representations of reality, thanothers. (2016a: 108)

Of course, if we are going to say that a causal modelmisrepresents the world, we must have a prior understandingof what the model represents. To misrepresent is to representinaccurately. This section is going to be focused on the question ofwhat it is for a causal model to represent accurately. There is ofcourse a further and subsequent question to be asked about when theinaccuracies of the model are negligible enough that the model may beappropriate orapt to use in a given context.

In one sense, this question should be seen as stipulative; the modelscome from humans, not gods. It is for us to say what they do and donot represent. Nonetheless, we should take care that our stipulationsdon’t undermine the purposes for which the models were designed,nor the use to which they are standardly put. Insofar as they aremeant to capture the notion ofinfluence which appears incausal claims like “how much I water my plant influences howtall it grows”, and insofar as they are meant to forgeconnections between influence and counterfactuals—or influenceand and token causation, or influence and control, or influence andchance—not just any stipulations will serve our purposes.Whether these connections are plausible or defensible will depend uponhow we understand the models; what we take them to be saying about theworld.

The formalism of the models is much the same, whether we arerepresenting token influence or type influence. But when it comes toexplaining what it takes for these models to be accurate, it willmatter whether we are talking about token influence or type influence.In the case of token influence, Hitchcock (2001a: 283–284)suggests that what it takes for a causal model to be correct is for acertain family of counterfactuals to be true:

A system of structural equations is an elegant means for representinga whole family of counterfactuals…The correctness of a set ofstructural equations…depends upon the truth of thesecounterfactuals.

On this kind of view, what it takes for a system of structuralequations to correctly represent a network of token influence is forsome corresponding counterfactuals to be true. There is a weaker and astronger version of this requirement. On the stronger version, werequire thatall of the (infinitely many) counterfactualswhich can be derived from the model (via the interventionistprocedure outlined in the previous subsection) are true. On the weakerversion, we require only the more limited class of counterfactualswhich say that, were any subset of the variables to have their valuessetvia an intervention, the equations for the non-intervenedupon variables would continue to be true. For illustration, take thefollowing system of equations,

The weak requirement is that, for any values \(x, y, z\), thefollowing counterfactuals hold:

One worry with the weak requirement is that the counterfactuals in (1)fail to capture the modularity of the structural equations. Recall:it’s important to distinguish between thestructuralequations \(Y \coloneqq X\) and \(Z \coloneqq X + Y\) and theequations \(Y = X\) and \(Z=X+Y\). Modularity is not just theclaim that, if we were to intervene to setX equal tox, the equations \(Y = X\) and \(Z=X+Y\) would continue to betrue. It is the claim that, if we were to intervene to setXequal tox, thestructural equations \(Y \coloneqqX\) and \(Z \coloneqq X + Y\) would continue tohold. Thisrequires, at least, that the counterfactuals

would still be true, for any values \(x'\) and \(y\), were weto intervene to setX equal tox. So it requiresthat the nested counterfactuals

are actually true. (And, indeed, these nested counterfactuals followfrom the system of equations, on the causal modeling semantics.)However, on many semantics for counterfactuals, this nestedcounterfactual does not follow from the counterfactuals given in(3.4.1) above. We might attempt to reduce nested counterfactuals likethese to counterfactuals with conjunctive antecedents by appeal to aprinciple of exportation, according to which \(\phi \boxto (\psi\boxto \chi)\) follows from \((\phi \wedge \psi) \boxto \chi\).However, on the causal modeling semantics, exportation is not valid ingeneral. Take the causal model above. Assuming counterfactuals withnecessarily false antecedents are vacuously true,

will be deemed true so long as \(x \neq x'\). But

will be deemed false. (See Briggs 2012, for related discussion, andGallow 2016, for an alternative approach.)

Some think that more than a family of true counterfactuals is requiredin order for a token causal model to be correct—or, at least, inorder for a causal model to provide us with a guide to token causation(see§3.3). Many, persuaded by the considerations from§1.2.3 above, include information about which variable values are more orless normal than which others. And Handfield et al. (2008) suggestthat the models must only represent relations of influence whichcorrespond to “connecting processes”, in the sense of Fair(1979), Salmon (1984, 1994), or Dowe (2000). The idea is this: if avariable,X, shows up on the right-hand-side ofY’s structural equation, then there must be somepossible state of the system (some possible assignment of values tothe variables in the model) such that, in that state, there is aconnecting process leading fromX’s value toY’s value.

Woodward (2003) focuses on systems oftype influence. Heproposes a series of non-reductive definitions which jointlycharacterize what it is for one type of variable,X, todirectly influence another type of variable,Y, relative to aset of variables types, \(\mathbf{V}\), which contains bothXandY. First, we are told:

X directly influencesY, relative to a set ofvariables \(\mathbf{V}\), iff there is a possible intervention onX which will changeY when all other variables in\(\mathbf{V}\) are held fixed at some value by interventions.

The definition makes reference to the notion of anintervention. (Notice: though Woodward is explicating therelation oftype influence, any particular intervention willbe atoken occurrence.) We saw how interventions are formallymodeled in the previous subsection; but Woodward gives the followingdefinition of an intervention in terms of aninterventionvariable. We are told:

I’s assuming some valueI=i is an interventiononX (with respect toY) iffI is anintervention variable forX (with respect toY) andI=i is a token cause of the value taken on byX.

This definition relies upon the notion of aninterventionvariable (forX with respect toY). We aretold:

I is an intervention variable forX with respect toY, relative to \(\mathbf{V}\), iff:

1. I influencesX;
2. there are values ofI such that, whenI takes onthose values,X’s value does not change when any of theother variables in \(\mathbf{V}\) which influenceX changetheir values; instead, the value ofX is determined by thevalue ofI alone;
3. every directed path of influence leading fromI toY travels throughX—that is, if there’ssome collection of variables \(Z_1, Z_2, \dots, Z_N \in \mathbf{V}\)such that \(I\) directly influences \(Z_1\), \(Z_1\) directlyinfluences \(Z_2\), \(\dots\), and \(Z_N \) directly influences \(Y\),then \(X = Z_i\), for some \(i \in \{ 1, 2, \dots, N \}\)
4. I is statistically independent of any variableZwhich directly influencesY and which is on a directed pathof influence which does not go throughX.

This definition appeals to the notion of (type) influence; so it willnot allow us to gives a reductive analysis of influence. But noticethat it doesn’t appeal to influence betweenX andY—and that is the relation which Woodward is attemptingto characterize. So while the series of definitions are not reductive,they are notviciously circular. They tell us something abouthow influence betweenX andY relates to influencebetween variables besidesX andY. (See the entry oncausation and manipulability for more.)

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Other Internet Resources

• Menzies, Peter, 2006, “A Structural Equations Account of Negative Causation”, PhilSci-Archive, deposited 10 October 2006.
• Weslake, Brad,unpublished manuscript.
• Causality, entry in Wikipedia.
• Causation, Laws, etc., category at PhilPapers.
• Schaffer, Jonathan, “The Metaphysics of Causation”,Stanford Encyclopedia of Philosophy (Sprin 2022 Edition),Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/spr2022/entries/causation-metaphysics/>. [This was the previous entry on this topic in theStanfordEncyclopedia of Philosophy — see theversion history.]

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Acknowledgments

Thanks to Phillip Bricker, Christopher Read Hitchcock, Doug Kutach,Laurie Paul, and Ignacio Silva.