Counterfactual Theories of Causation (Stanford Encyclopedia of Philosophy) Cite this entry Search the SEP • Advanced Search • Tools • RSS FeedTable of Contents• What's New• Archives• Projected ContentsEditorial Information• About the SEP• Editorial Board• How to Cite the SEP• Special CharactersSupport the SEPContact the SEP ©Metaphysics Research Lab,CSLI,Stanford University  Open access to the SEP is made possible by a world-wide funding initiative. Please Read How You Can Help Keep the Encyclopedia FreeCounterfactual Theories of CausationFirst published Wed Jan 10, 2001; substantive revision Sun Mar 30, 2008The basic idea of counterfactual theories of causation is that themeaning of causal claims can be explained in terms ofcounterfactual conditionals of the form “If A had notoccurred, C would not have occurred”. While counterfactual analyses have been given of type-causal concepts, most counterfactual analyses have focused on singular causal or token-causal claims of the form “event c caused event e”. Analyses of token-causation have become popular in the last thirty years, especially since the development in the 1970's of possible world semantics for counterfactuals. The best known counterfactualanalysis of causation is David Lewis's (1973b) theory. However,intense discussion over thirty years has cast doubt on the adequacyof any simple analysis of singular causation in terms ofcounterfactuals. Recent years have seen a proliferation of differentrefinements of the basic idea to achieve a closer match withcommonsense judgements about causation. 1. Early Counterfactual Theories2. Lewis's 1973 Counterfactual Analysis 2.1 Counterfactuals and Causal Dependence 2.2 The Temporal Asymmetry of Causal Dependence 2.3 Transitivity and Preemption 2.4 Chancy Causation 2.5 The Theory's Advantages 3. Problems for Lewis's Counterfactual Theory 3.1 Context-sensitivity 3.2 Temporal Asymmetry 3.3 Transitivity 3.4 Preemption 4. Later Developments 4.1 Lewis's 2000 Theory 4.2 Causation as Intrinsic Relation 4.3 The Structural Equations Framework BibliographyOther Internet ResourcesRelated Entries1. Early Counterfactual TheoriesThe first explicit definition of causation in terms ofcounterfactuals was, surprisingly enough, given by Hume, when hewrote: “We may define a cause to be an object followed by another,and where all the objects, similar to the first, are followed byobjects similar to the second. Or, in other words, where, ifthe first object had not been, the second never had existed.”(1748, Section VII). It is difficult to understand how Hume couldhave confused the first, regularity definition with the second, verydifferent counterfactual definition. At any rate, Hume never explored the alternative counterfactualapproach to causation. In this, as in much else, he was followed bygenerations of empiricist philosophers. The chief obstacle inempiricists' minds to explaining causation in terms ofcounterfactuals was the obscurity of counterfactuals themselves,owing chiefly to their reference to unactualisedpossibilities. Starting with J. S. Mill (1843), empiricists tried toanalyse counterfactuals ‘metalinguistically’ in terms ofimplication relations between statements. The rough idea is that acounterfactual of the form “If it had been the case that A, it wouldhave been the case that C” is true if and only if there is anauxiliary set S of true statements consistent with theantecedent A, such that the members of S, whenconjoined with A, imply the consequent C. Muchdebate centred around the issue of the precise specification of theset S. (See N. Goodman 1947.) Most empiricists agreed thatS would have to include statements of laws of nature, whilesome thought that it would have to include statements of singularcausation. While the truth conditions of counterfactuals remainedobscure in these ways, few empiricists thought it worthwhile to tryto explain causation via counterfactuals. Indeed, the first real attempts to present rigorous counterfactualanalyses of causation came only in the late 1960's. (See A. Lyon1967.) Typical of these attempts was J. L. Mackie's counterfactualanalysis in Chapter 2 of his seminal book The Cement of theUniverse (1974). As well as offering a sophisticated regularitytheory of causation ‘in the objects’, Mackie presented acounterfactual account of the concept of a cause as “what makes thedifference in relation to some background or causal field” (1980,p.xi). Mackie's account of the concept of causation is rich ininsights, especially concerning its relativity to a field ofbackground conditions. However, his account never gained as muchattention as his regularity theory of causation ‘in theobjects’, no doubt because his view of counterfactuals (in his(1973)), as condensed arguments that do not have truth values,compounded empiricists' scepticism about counterfactuals. The true potential of the counterfactual approach to causation didnot become clear until counterfactuals became better understoodthrough the development of possible world semantics in the early1970's.2. Lewis's 1973 Counterfactual AnalysisThe best known and most thoroughly elaborated counterfactual theory ofcausation is David Lewis's theory in his (1973b), which was refinedand extended in articles subsequently collected in his (1986a). Inresponse to doubts about the theory's treatment of preemption, Lewissubsequently proposed a fairly radical revision of the theory. (Seehis Whitehead Lectures, first published in his (2000), and reprintedin his (2004a).) In this section we shall confine our attention to theoriginal 1973 theory, deferring the later changes he proposed forconsideration below. 2.1 Counterfactuals and Causal Dependence 2.2 The Asymmetry of Causal Dependence 2.3 Preemption and Transitivity 2.4 Chancy Causation 2.5 The Theory's Advantages2.1 Counterfactuals and Causal DependenceLike most contemporary counterfactual theories, Lewis's theory employsa possible world semantics for counterfactuals. Such a semanticsstates truth conditions for counterfactuals in terms of similarityrelations between possible worlds. Lewis famously espouses a realismabout possible worlds, according to which non-actual possible worldsare real concrete entities on a par with the actual world. (SeeLewis's defence of modal realism in his (1986e).) However, mostcontemporary philosophers would seek to deploy the explanatorilyfruitful possible worlds framework while distancing themselves fromfull-blown realism about possible worlds themselves. For example, manywould propose to understand possible worlds as maximally consistentsets of propositions; or even to treat them instrumentally as usefultheoretical entities having no independent reality. The central notion of a possible world semantics for counterfactualsis a relation of comparative similarity between worlds (Lewis1973a). One world is said to be closer to actuality thananother if the first resembles the actual world more than the seconddoes. Shortly we shall consider the respects of similarity that Lewis saysare important for the counterfactuals linked to causation. For now wesimply note two formal constraints he imposes on this similarityrelation. First, the relation of similarity produces a weak orderingof worlds so that any two worlds can be ordered with respect to theircloseness to the actual world, with allowance being made for ties incloseness. Secondly, the actual world is closest to actuality,resembling itself more than any other world resembles it. In terms of this similarity relation, the truth condition for thecounterfactual “If A were (or had been) the case, Cwould be (or have been) the case”, is stated as follows:(1)“If A were the case, C would be the case” is true in the actual worldif and only if (i) there are no possible A-worlds; or (ii)some A-world where C holds is closer to theactual world than is any A-world where C does nothold. We shall ignore the first case in which the counterfactual isvacuously true. The fundamental idea of this analysis is that thecounterfactual “If A were the case, C would be thecase” is true just in case it takes less of a departure from actualityto make the antecedent true along with the consequent than to make theantecedent true without the consequent. In terms of counterfactuals, Lewis defines a notion of causaldependence between events, which plays a central role in his theory(1973b).(2)Where c and e are two distinct possibleevents, e causally depends on c if and only if, ifc were to occur e would occur; and if c were not tooccur e would not occur.This condition states that whether e occurs or not depends onwhether c occurs or not. Where c and e areactual occurrent events, this truth condition can be simplifiedsomewhat. For in this case it follows from the second formal conditionon the comparative similarity relation that the counterfactual “Ifc were to occur e would occur” is automaticallytrue: this formal condition implies that a counterfactual with trueantecedent and true consequent is itself true. Consequently, the truthcondition for causal dependence becomes:(3)Where c and e are two distinct actual events,e causally depends on c if and only if, if cwere not to occur e would not occur. The right hand side of this condition is, of course, Hume's seconddefinition of causation. (As we shall see shortly, Lewis's officialdefinition of causation differs from it, as he defines causation notin terms of causal dependence directly, but in terms of chains ofcausal dependence.) Why is it plausible to think that causation isconceptually linked with counterfactuals in the way specified by thisdefinition of causal dependence? One reason is that the idea of acause is conceptually linked with the idea of something that makes adifference and this idea in turn is best understood in terms ofcounterfactuals. In Lewis's words: “We think of a cause as somethingthat makes a difference, and the difference it makes must be adifference from what would have happened without it. Had it beenabsent, its effects — some of them, at least, and usually all— would have been absent as well.” (1973b, p.161)There are three important things to note about the definition ofcausal dependence. First, it takes the primary relata of causaldependence to be events. Lewis's own theory of events(1986b) construes events as classes of possible spatiotemporalregions. However, very different conceptions of events are compatiblewith the basic definition. Indeed, it even seems possible toformulate it in terms of facts rather than events. (For instance, seeMellor 1996, 2004.) Secondly, the definition requires the causally dependent events to bedistinct from each other. Distinctness means that the eventsare not identical, neither overlaps the other, and neither implies theother. This qualification is important if spurious non-causaldependences are to be ruled out. (For this point see Kim 1973 andLewis 1986b.) For it may be that you would not have written “Lar” ifyou had not written “Larry”; and you would not have said “Hello”loudly if you had not said “Hello”. But neither dependence counts as acausal dependence since the paired events are not distinct from eachother in the required sense.Thirdly, the counterfactuals that are employed in the analysis are tobe understood according to what Lewis calls the standardinterpretation. There are several possible ways of interpretingcounterfactuals; and some interpretations give rise to spuriousnon-causal dependences between events. For example, suppose that theevents c and e are effects of a common caused. It is tempting to reason that there must be a causaldependence between c and e by engaging in thefollowing piece of counterfactual reasoning: if c had notoccurred, then it would have to have been the case that d didnot occur, in which case e would not have occurred. But Lewissays these counterfactuals, which he calls backtrackingcounterfactuals, are not to be used in the assessment of causaldependence. The right counterfactuals to be used arenon-backtracking counterfactuals that typically hold the pastfixed up until the time at which the counterfactual antecedent issupposed to obtain.2.2 The Temporal Asymmetry of Causal DependenceWhat constitutes the direction of the causal relation? Why is thisdirection typically aligned with the temporal direction from past tofuture? In answer to these questions, Lewis (1979) argues that thedirection of causation is the direction of causal dependence; and itis typically true that events causally depend on earlier events butnot on later events. He emphasises the contingency of the latter factbecause he regards backwards or time-reversed causation as aconceptual possibility that cannot be ruled out apriori. Accordingly, he dismisses any analysis of counterfactualsthat would deliver the temporal asymmetry by conceptual fiat. Lewis's explanation of the temporal asymmetry of counterfactualdependence is based on a de facto asymmetry about the actualworld. He defines a determinant for an event as any set ofconditions jointly sufficient, given the laws of nature, for theevent's occurrence. (Determinants of an event may be causes or tracesof the event.) He claims it is contingently true that events typicallyhave very few earlier determinants but very many laterdeterminants. As an illustration, he cites Popper's (1956) example ofa spherical wavefront expanding outwards from a point source. This isa process where each sample of the wave postdetermines what happens atthe point at which the wave is emitted. He says the reverse process inwhich a spherical wave contracts inward with each sample of wavepredetermining what happens at the point the wave is absorbed wouldobey the laws of nature but seldom happens in actual fact. Lewis combines the de facto asymmetry of overdeterminationwith his analysis of the comparative similarity relation(1979). According to this analysis, there are several respects ofsimilarity to be taken into account in evaluating non-backtrackingcounterfactuals: similarity with respect to laws of nature and alsosimilarity with respect to particular matters of fact. Worlds are moresimilar to the actual world the fewer miracles or violations of theactual laws of nature they contain. Again, worlds are more similar tothe actual world the greater the spatio-temporal region of perfectmatch of particular fact they have with the actual world. If theactual world is governed by deterministic laws, these rules will clashin assessing which counterfactual worlds are more similar to theactual world. For a world that makes a counterfactual antecedent truemust differ from the actual world either in allowing some violation ofthe actual laws, or in differing from the actual world in particularmatters of fact. Lewis's analysis allows a tradeoff between thesecompeting respects of similarity in such cases. It implies that worldswith an extensive region of perfect match of particular fact can beconsidered very similar to the actual world provided that the match inparticular facts with the actual world is achieved at the cost of asmall, local miracle, but not at the cost of a big, diversemiracle. Taken by itself, this account contains no built-in timeasymmetry. That comes only when it is combined with the asymmetry ofoverdetermination. To see how the two parts combine, consider the famous example ofNixon and the Nuclear Holocaust. An early objection to Lewis'saccount of counterfactuals (Fine 1975) was that,counterintuitively, it makes this counterfactual false:(4) If Nixon had pressed the button, there would have been a nuclearwar. The argument is that a world in which Nixon pressed the button, butsome minute violation of the laws then prevented a nuclear war, ismuch more like the actual world than one in which Nixon pressed thebutton and a nuclear war took place. Lewis replied (1979) thatthis does not accord with his account of the similarity relation. Onthis account, a button-pressing world that diverges from the actualworld by virtue of a miracle is more like the actual world than abutton-pressing world that converges with the actual world by virtueof a miracle. For in view of the asymmetry of overdetermination, thedivergence miracle that allows Nixon to press the button need only bea small, local miracle, but the convergence miracle required to wipeout the traces of Nixon's pressing the button must be a verybig, diverse miracle. Of course, if the asymmetry ofoverdetermination went in the opposite temporal direction, the verysame standards of similarity would dictate the opposite verdict. In general, then, the symmetric analysis of similarity, combined withthe de facto asymmetry of overdetermination, implies thatworlds that accommodate counterfactual changes by preserving theactual past and allowing for divergence miracles are more similar tothe actual world than worlds that accommodate such changes by allowingfor convergence miracles that preserve the actual future. This fact inturn implies that, where the asymmetry of overdetermination obtains,the present counterfactually depends on the past, but not on thefuture.2.3 Transitivity and PreemptionAccording to Lewis, causal dependence between actual events issufficient for causation, but not necessary (1973b): it is possible tohave causation without causal dependence. This can happen in thefollowing way. Suppose that c causes d in virtue ofthe fact that d causally depends on c, andd causes e in virtue of the fact that ecausally depends on d. Then because causation is transitive,Lewis insists, c must cause e. However, becausecausal dependence is not transitive like causation, the causalrelation between c and e may not be matched by acausal dependence. (We shall shortly consider an example of thiskind.) To overcome this problem Lewis extends causal dependence to atransitive relation by taking its ancestral. He defines a causalchain as a finite sequence of actual events c,d, e,… where d causally depends onc, e on d, and so on throughout thesequence. Then causation is finally defined in these terms:(5)c is a cause of e if and only if there existsa causal chain leading from c to e. This definition not only ensures the transitivity of causation, butit also appears to solve an additional problem to do with preemptionthat is illustrated by the following example. Suppose that two crackmarksmen conspire to assassinate a hated dictator, agreeing that oneor other will shoot the dictator on a public occasion. Actingside-by-side, assassins A and B find a good vantagepoint, and, when the dictator appears, both take aim. A pullshis trigger and fires a shot that hits its mark, but Bdesists from firing when he sees A pull his trigger. Hereassassin A's actions are the actual cause of the dictator'sdeath, while B's actions are a preempted potentialcause. (Lewis distinguishes such cases of preemption from casesof symmetrical overdetermination in which two processesterminate in the effect, with neither process preempting theother. Lewis believes that these cases are not suitable test cases fora theory of causation since they do not elicit clear judgements.) Theproblem raised by this example of preemption is that both actions areon a par from the point of view of causal dependence: if neitherA nor B acted, then the dictator would not havedied; and if either had acted without the other, the dictator wouldhave died. However, given the definition of causation in terms of causal chains,Lewis is able to distinguish the preempting actual cause from thepreempted potential cause. There is a causal chain running fromA's actions to the dictator's death, but no such chainrunning from B's actions to the dictator's death. Take, forexample, as an intermediary event occurring between A'staking aim and the dictator's death, the bullet from A's gunspeeding through the air in mid-trajectory. The speeding bulletcausally depends on A's action since the bullett would nothave been in mid-trajectory without A's action; and thedictator's death causally depends on the speeding bullett since by thetime the bullett is in mid-trajectory B has refrained fromfiring so that the dictator would not have died without the presenceof the speeding bullett. (Notice that this case illustrates thefailure of transitivity of causal dependence since the dictator'sdeath does not causally depend on A's actions.) Hence, wehave a causal chain, and so causation. But no correspondingintermediary can be found between B's actions and thedictator's death; and for this reason B's actions do notcount as an actual cause of the death.2.4 Chancy CausationSo far we have considered how the counterfactual theory of causationworks under the assumption of determinism. But what about causationwhen determinism fails? Lewis (1986c) argues that chancy causation isa conceptual possibility that must be accommodated by a theory ofcausation. Indeed, contemporary physics tells us the actual worldabounds with probabilistic processes that are causal in character. Totake a familiar example (Lewis 1986c): suppose that you mischievously hookup a bomb to a radioactive source and geiger counter in such a waythat the bomb explodes when the counter registers a certain number ofclicks. If it happens that the counter registers the required numberof clicks and the bomb explodes, your act caused the explosion, eventhough there is no deterministic connection between them. In order to accommodate chancy causation, Lewis (1986c) defines amore general notion of causal dependence in terms of chancycounterfactuals. These counterfactuals are of the form “If Awere the case Pr (C) would be x”, where thecounterfactual is an ordinary would-counterfactual, interpretedaccording to the semantics above, and the Pr operator is aprobability operator with narrow scope confined to the consequent ofthe counterfactual. Lewis interprets the probabilities involved astemporally indexed single-case chances. (See his (1980) for the theoryof single-case chance.) The more general notion of causal dependence reads:(6)Where c and e are distinct actual events, ecausally depends on c if and only if, if c were notoccurred, the chance of e's occurring would be much less than its actual chance.This definition covers cases of deterministic causation in which thechance of the effect with the cause is 1 and the chance of the effectwithout the cause is 0. But it also allows for cases of irreducibleprobabilistic causation where these chances can take non-extremevalues. It is similar to the central notion of probabilistic relevanceused in probabilistic theories of type-causation, except that itemploys chancy counterfactuals rather than conditionalprobabilities. (See the discussion in Lewis 1986c for the advantagesof the counterfactual approach over the probabilistic one. Also seethe entry “Probabilistic Causation”.) The rest of the theory of chancy causation follows the outlines ofthe theory of deterministic causation. Causal dependence is extendedto a transitive notion by taking its ancestral. As before, we havecausation when we have one or more steps of causal dependence.2.5 The Theory's AdvantagesBefore turning to survey some of the problems confronting Lewis'stheory of causation, it is worthwhile pausing to consider some of theadvantages it affords.At the time that Lewis advanced his original theory, regularitytheories of causation were the orthodoxy. Taking Hume's firstdefinition as their point of departure, these theories definedcausation in terms of subsumption under lawful regularities. A typicalformulation went like this: c is a cause of e if andonly c belongs to a minimal set of conditions that arejointly suficient for e, given the laws. It was well knownthat theories of this kind were faced with a number of recalcitrantcounterexamples. Thus, while c might belong to a minimal setof sufficient conditions for e when c is a genuinecause of e, this might also be true when c is aneffect of e — an effect which could not have occurred,given the laws and the actual circumstances, except by being caused bye. Or it might be true when c and e arejoint effects of a common deterministic cause. Or when c is apreempted potential cause of e — something that did notcause e, but would have done so if the actual cause had beenabsent.In contrast, Lewis's counterfactual analysis of causation is notsubject to the same counterexamples, so long the counterfactuals inthe definition of causal dependence and causation are interpreted in anon-backtracking fashion. The theory implies that even if cbelongs to a minimal set of sufficient conditions for e,e will not causally depend on c when coccurs after e as its effect, since earlier events do nottypically causally depend on later events. Nor will ecausally depend on c when c and e are jointeffects of a common cause, since the non-backtracking counterfactual“If c had not occurred, e would still have occurred”will be true in view of the fact that it holds fixed the presence ofthe common cause. Nor will c count as a cause of ewhen c is a preempted potential cause of e in atypical case of preemption. For, as we have seen, c will notbe connected to e by a chain of causal dependences.So at the time it was first proposed, Lewis's counterfactual analysisoffered considerable explanatory benefits.3. Problems for Lewis's Counterfactual TheoryIn this section we consider the principal difficulties forLewis's theory that have emerged in discussion over the lastthirty years.3.1 Context-sensitivity3.2 Temporal Asymmetry3.3 Transitivity3.4 Preemption3.1 Context-sensitivityOne relatively overlooked aspect of the concept of causation is itssensitivity to contextual factors. In so far as Lewis's theoryoverlooks this context-sensitivity, it represents a problem for thetheory. The theory assumes that causation is an absolute relation whosenature does not vary from one context to another. (This follows fromthe way the counterfactuals that define the central notion of causaldependence are governed by a unique, context-invariant system ofweighted respects of similarity.) According to the theory, any eventbut for which an effect would not have occurred is one of the effect'scauses. But this generates some absurd results. For example, suppose acamper lights a fire, a sudden gust of wind fans the fire, the firegets out of control and the forest burns down. It is true that if thecamper had not lit the fire, the forest fire would not haveoccurred. But it is also true that the forest fire would not haveoccurred if any of a vast number of contingencies, including thecamper's birth and his failure to be struck down by a meteor beforestriking the match, had not occurred. But commonsense draws adistinction between causes and background conditions, ranking thecamper's lighting of the fire among the former, and his birth and hisfailure to be struck down by a meteor, among the latter. H. L. A. Hart and A. Honoré (1965; 2nd ed 1985) argue that thedistinction between causes and conditions is relative to context in atleast two different ways. One form of relativity might be calledrelativity to the context of occurrence. If a forest is destroyed byfire, the presence of oxygen would be cited as a mere condition of theforest's destruction. On the other hand, if a fire breaks out in alaboratory where oxygen is deliberately excluded, it may beappropriate to cite the presence of oxygen as a cause of the fire. Thesecond form of relativity might be called relativity to the context ofenquiry. For example, the cause of a great famine in India may beidentified by an Indian farmer as the drought, but the World FoodAuthority may identify the Indian government's failure to build upreserves as the cause, and the drought as a mere condition. For the most part, Lewis ignores these subtle context-sensitivedistinctions, as he says he is interested in a broad notion ofcause. In his view (1986d), every event has an objective causalhistory consisting of a vast structure of events ordered by causaldependence. The human mind may select parts of the causal history forattention, perhaps different parts for different purposes ofenquiry. However, Lewis does not specify the ‘principles ofinvidious selection’ by which some parts of the causal historyare selected for attention, except to mention the relevance of Grice'smaxims of conversation. But Grice's maxims of conversation, as generalprinciples of rational information exchange, are not well suited toexplaining the causation-specific distinctions we draw. As severalphilosophers have pointed out (A. Garfinkel (1981); C. Hitchcock(1996a, 1996b); P. Lipton (1990); J. Woodward (1984); and B. VanFraassen (1981)), some of the contextual principles behind our causaljudgements seem to rely on considerations concerning which class ofsituations the effect is contrasted with. Thus, in the example of the Indian famine, we contrast the actualsituation in which a famine occurs with another situation in whichnormal conditions prevail and a famine does not occur. A cause is thenthought of as a factor that makes the difference between thesesituations; and the background conditions are thought of as thosefactors that are common to the two situations. In different contextsof enquiry, the contrast situation is framed in different terms. Afarmer may take the contrast situation to be the normal situation inwhich the government does not stockpile food reserves but there is nofamine. In this case it would be reasonable for the farmer to identifythe drought as the factor that makes the difference between thiscontrast situation and the actual situation in which there isfamine. On the other hand, an official of the World Food Authoritywith a different conception of what normally happens may take thecontrast situation to be one in which governments build up foodreserves as a precaution against droughts. Consequently, it would bereasonable for the official to see the failure of the government tobuild up food reserves as the factor that makes the difference betweenthe contrast situation and the actual situation in which there is afamine. (For discussion of the relevance of contrastive explanation tothe causes/conditions distinction see Menzies 2004a; 2007.) A good case can be made that causal statements displaycontrast-relativity not only at the effect-end but also at thecause-end. (See Hitchcock, 1996a, 1996b; Maslen 2004; Schaffer 2005)Recognising this helps to deal with a problem affecting Lewis'soriginal theory. In evaluating whether an event c caused anevent e, Lewis's theory says we have to consider what wouldhave happened in those closest worlds in which c did notoccur. For example, in evaluating whether the camper's lighting of thefire caused the forest, we have to consider what would have happenedin those closest possible worlds in which the camper's action oflighting the fire did not occur. Are these worlds in which the camperdoes not light the fire but does something else instead, or are theyworlds in which he lights the fire in slightly different manner(perhaps with a lighter instead of matches) or at a slightly differenttime (three minutes later when the wind died down)? In order to answersuch questions, Lewis says it is necessary to say how much of a changeor a delay it takes for an event to become an altogether differentevent, rather than a different version of the same event. (Lewissometimes discusses this issue as the question of how fragile eventsare: a modally fragile event is one which cannot occur in adifferent manner or at a different time from its actual manner andtime of occurrence. See Lewis 1986b.) The problem, as he sees it, isthat there is no unique principled way in which we do this: there islinguistic indeterminancy about what event nominals refer to. Hewrites: “We have not made up our minds: and if we presuppose sometimesone answer and sometimes another answer, we are entirely within ourlinguistic rights. This is itself a big problem for a counterfactualanalysis of causation, quite apart from the problem of preemption.”(2000, p.186) However, if we recognise that the cause-end of causal statementsdisplays contrast-relativity as well as the effect-end, we can obviatethe need to provide an account of the identity of events undercounterfactual changes. For example, suppose we are interested in whythe forest fire took one path P1 rather thananother path P2. Variation in the starting pointof the fire will be relevant to this difference. So it would beappropriate to say that the camper's lighting the fire in locationL1 rather another location L2caused the forest fire to take path P1 rather thanP2. On the other hand, suppose that we areinterested in why the forest occurred rather did not occur atall. Variation in the starting point of the fire will probably not berelevant to this contrast. Rather the appropriate causal statementwill be one that says the camper's lighting the fire (in some or otherlocation) rather his not lighting it (in any location) caused the fireto occur rather than not to occur. Such causal statements reveal therelevant contrasts at both the cause- and the effect-ends. Sometimes,such contrasts are indicated by the use of emphasis as in “The fire'sstarting in location L1 caused the fire to takepath P1”. But more often than not the surface formof causal statements does not disclose the contrasts that are intendedand they must be supplied by context. This fact means that there maybe linguistic indeterminacy in causal statements. But it is notindeterminacy about the reference of event nominals, but rather aboutthe situations that are intended as contrasts for the cause and theeffect. Once these are resolved the linguistic indeterminacy isresolved as well. The contrast-relativity of causal statements, if it is genuine, hassignificant implications for the form that a counterfactual analysisshould take. Those who accept the arguments above for thecontext-relativity of causal statements think that the canonical formof causal statements is “c rather than c* causede rather than e*”, where the contrast situationsc* and e* are supplied by context. This suggeststhat the definition of causal dependence should not be formulated interms of the counterfactual “If c had not occurred,e would not have occurred”, but the more specificcounterfactual “If c* had occurred instead of c,then e* would have occurred instead of e”. Thisformulation has several advantages over the old formulation. (SeeSchaffer 2005.) Its chief advantage from the point of our discussionis that it obviates the need for the counterfactual theory to providean account of the identity of events under hypothetical changes. Withthis new formulation, there is no need to work out whether c*and e* are identical with, or different from, c ande,respectively. It is simply stipulated on the basis ofcontextual considerations that c* and e* areintended to act as contrasts to c and e.3.2 Temporal AsymmetryThere have been several important critical discussions of Lewis'sexplanation of the temporal asymmetry of causation. (See A. Elga2000;M. Frisch 2005; D. Hausman 1998, Chap. 6;P. Horwich 1987,Chap. 10; and H. Price 1996, Chap. 6.) One kind of criticism has focused on the psychological implausibilityof Lewis's explanation. (See Horwich 1987.) Recall that theexplanation appeals, on the one hand, to a system of weighted respectsof similarity between possible worlds that is delivered by apriori conceptual analysis and, on the other hand, to an asymmetryof overdetermination that is claimed to be a contingent aposterioritruth about the actual world. The two-part explanationis supposed to employ facts that are sufficiently well known to play arole in the explanation of our linguistic use ofcounterfactuals. However, it is psychologically implausible that theintricate system of weighted respects of similarity involvingcomparison of miracles of different sizes could capture the intuitivesimilarity relation used in counterfactual reasoning. Why should wehave developed such a baroque notion of similarity? Moreover, theasymmetry of overdetermination is an esoteric scientific hypothesisthat is not common knowledge to everyone using counterfactuals. So itis very unlikely that this hypothesis could account for ordinaryspeakers' mastery of the temporal asymmetry of counterfactuals. (ForLewis's reply to this criticism see Postscript E to “CounterfactualDependence and Time's Arrow” in his (1986a, p. 66).) Another criticism is that the asymmetry of overdetermination does notexist in the form required to support Lewis's explanation of thetemporal asymmetry of counterfactuals. Lewis's idea is that any evente has many postdeterminants and few predeterminants, where apredeterminant or postdeterminant of an event is a set of conditionsthat are jointly sufficient, given the laws of nature, for theoccurrence of the event. But if Lewis is assuming that the lawsinvolved are like those of classical mechanics, he is mistaken on thisscore. For a theory that is time symmetric and deterministic in boththe forward and backward direction will imply that for any local evente and any time t, there is a unique set ofconditions obtaining at t that are necessary and sufficient,given the laws, for the occurrence of the event e. Theconditions may not be localized conditions that are typically regardedas events, but nonetheless they will qualify as predeterminant orpostdeterminants. For example, consider Popper's example of the wavespreading out from a point source. If there is a process thatpostdetermines what happens at the point at which the wave is emitted,there is also a process, perhaps a very unlocalized process, thatpredetermines this. Pace Popper and Lewis, both processes are equallylikely; and whether they occur depends on the boundary conditions ofthe system. (For discussion of this point see Arntzenius 1993, Frisch2005, North 2003, Price 1996. Also see the entry “ThermodynamicAsymmetry in Time”.)A related criticism concerns the asymmetry of miracles that is centralto Lewis's account of the temporal asymmetry of causation. Theasymmetry of miracles consists in the fact that a miracle thatrealises a counterfactual antecedent about particular facts at timet by having a possible world diverge from the actual worldjust before the time t is smaller and less diverse than amiracle that realises the same counterfactual antecedent and makes apossible world converge to the actual world after the timet. Adam Elga (2000) has argued that the asymmetry of miraclesdoes not hold in many cases.Elga's argument proceeds by way of an example: Gretta cracks an egginto a hot frying pan at 8:00am and at 8:05am the egg iscooked. Consider the process that occurs in the period from 8:00am to8:05am, run backwards in time: a cooked egg sits in the frypan; itcoalesces into a raw egg and leaps upward; and a shell closes aroundit. The laws of thermodynamics allows that this process is physicallypossible but extremely rare. These laws also state that the process isvery sensitive in its initial conditions: even the slightest changesin the molecules making up the state of the cooked egg would result inthe process evolving in such a way that the cooked egg continues tosit in the pan rather than coalescing into a raw egg and leapingupwards. But this is, Elga points out, exactly the kind of change thatwould make for a “convergence miracle”. Take the state of the actualworld at 8:05am, holding fixed its future after this point; make somesmall changes to the molecules making up this state; and then run thelaws of thermodynamics backwards in time, and we will almost certainlyarrive at a state in which the egg sits in the pan growingcolder. This state will be one in which Gretta does not crack theegg. The small change in the state of the actual world at 8:05am is a“convergence miracle” that yields a possible world that realises thecounterfactual proposition that Gretta does not crack the egg at8:00am while holding fixed the actual future after 8:05am. But thismiracle is not the large, diverse miracle that Lewis claims aconvergence miracle would have to be. 3.3 TransitivityAs we have seen, Lewis builds transitivity into causation by definingit in terms of chains of causal dependence. The transitivity ofcausation fits with some of our explanatory practices. For example,historians wishing to explain some significant historical event willtrace the explanation back through a number of causal links,concluding that the event at the beginning of the causal chain isresponsible for the event being explained. On the other hand, a numberof counter-examples have been presented which cast doubt ontransitivity. (Lewis 2004a presents a short catalogue of thesecounterexamples.) Here is a sample of three counterexamples.First, an example due to Michael McDermott (1995). A andB each have a switch in front of them, which they can move tothe left or right. If both switches are thrown into the same position,a third person C receives a shock. A does not wantto shock C. Seeing B's switch in the left position,A moves her switch to the right. B does want toshock C. Seeing A's switch thrown to the right, shenow moves her switch to the right as well. C receives ashock. Clearly, A's throwing her switch to the right causesB to throw her switch to the right, which in turn causesC to receive the shock. But A attempted to preventthe shock so that it seems unreasonable to say that A's movecauses C to be shocked.Second, an example due to Ned Hall (2004). A person is walkingalong a mountain trail, when a boulder high above is dislodged andcomes careering down the mountain slopes. The walker notices theboulder and ducks at the appropriate time. The careering bouldercauses the walker to duck and this, in turn, causes his continuedstride. (This second causal link involves double prevention:the duck prevents the collision between walker and boulder which, hadit occurred, would have prevented the walker's continued stride.)However, the careering boulder is the sort of thing that would preventthe walker's continued stride and so it seems counterintuitive to saythat it causes the stride.Third, an example due to Douglas Ehring (1987). Jones puts somepotassium salts into a hot fire. Because potassium compounds produce apurple flame when heated, the flame changes to a purple colour, thougheverything else remains the same. The purple flame ignites someflammable material nearby. Here we judge that putting the potassiumsalts in the fire caused the purple flame, which in turn caused theflammable material to ignite. But it seems implausible to judge thatputting the potassium salts in the fire caused the flammable materialto ignite. Various replies have been made to these counterexamples. The lastcounterexample seems the most easily deflected. For example, Maslen(2004), who endorses the contrast-relativity of causl statements, hasargued that this example is misdiagnosed as a counterexample totransitivity, as the contrast situation at the effect-end of the firstcausal statement does not match up with the contrast situation thecause-end of the second causal statement. Thus, the first causalstatement should be interpreted as saying that Jones's puttingpotassium salts in the fire rather not doing so caused the flame to turn purple rather than yellow; but the second causalstatement should be interpreted as saying that the purple fire'soccurring rather than not occurring caused the flammable materialto ignite rather not to ignite. Where there is a mismatch of thiskind, we do not have a genuine counterexample to transitivity. L. Paul(2004) offers a similar diagnosis of the last example, though herdiagnosis proceeds in terms of event aspects, which she takes to becausation primary relata. She argues similarly that there is mismatchbetween the event aspect that is the effect of the first causal link(the flame's being a purple colour) and the event aspect that is thecause of the second causal link (the flame's touching the flammablematerial). The first and second examples cannot be handled in the same way. Somedefenders of transitivity have replied that our intuitions about theintransitivity of causation in these examples are misleading. Forinstance, Ned Hall (2000) has argued that we should suspect ourintuition in the second example because it involves double prevention,which he claims is not a genuine kind of causation. Thus, he deniesthat the walker's ducking caused his continued stride since this holdsonly by double prevention.(This also commits him to denying thatcausal dependence is sufficient for causation, since the walker'scontinued stride causally depends on his ducking the boulder.) Heoffers a different diagnosis of why our intuitions go awry in thefirst example. Lewis (2004a) adopts a similar strategy of trying toexplain away the force of our intuitions in these examples. He pointsout that the counterexamples to transitivity typically involve astructure in which a c-type event generally prevents ane-type but in the particular case the c-eventactually causes another event that counters the threat and causes thee-event. If we mix up questions of what is generallyconducive to what, with questions about what caused what in thisparticular case, he says, we may think that it is reasonable to denythat c causes e. But if we keep the focus sharply onthe particular case, we must insist that c does in fact causee.The debate about the transitivity of causation is not easily settled,partly because it is tied up with the issue of how it is best for acounterfactual theory to deal with examples of preemption. As we haveseen, Lewis's counterfactual theory relies on the transitivity ofcausation to handle cases of preemption. If such cases could behandled in some other way, that would take some of the theoreticalpressure off the theory, allowing it concede the persuasivecounterexamples to transitivity without succumbing to the difficultiesposed by preemption. (For more on this point see Hitchcock 2001.)3.4 PreemptionAs we have seen, Lewis employs his strategy of defining causation interms of chains of causal dependence not only to make causationtransitive, but also to deal with preemption examples. However, thereare preemption examples that this strategy cannot deal withsatisfactorily. Difficulties concerning preemption have proven to bethe biggest bugbear for Lewis's theory. In his (1986c), Lewis distinguishes cases of early andlate preemption. In early preemption examples, the processrunning from the preempted alternative is cut short before the mainprocess running from the preempting cause has gone to completion. Theexample of the two assassins, given above, is an example of thissort. The theory of causation in terms of chains of causal dependencecan handle this sort of example. In contrast, cases of latepreemption are ones in which the process running from the preemptedcause is cut short only after the main process has gone to completionand brought about the effect. The following is an example of latepreemption due to Hall (2004). Billy and Suzy throw rocks at a bottle. Suzy throws first so thather rock arrives first and shatters the glass. Without Suzy'sthrow, Billy's throw would have shattered the bottle. However,Suzy's throw is the actual cause of the shattered bottle, whileBilly's throw is merely a preempted potential cause. This is acase of late preemption because the alternative process (Billy'sthrow) is cut short after the main process (Suzy's throw) hasactually brought about the effect. Lewis's theory cannot explain the judgement that Suzy'sthrow was the actual cause of the shattering of the bottle. For thereis no causal dependence between Suzy's throw and the shattering,since even if Suzy had not thrown her rock, the bottle would haveshattered due to Billy's throw. Nor is there a chain of stepwisedependences running cause to effect, because there is no eventintermediate between Suzy's throw and the shattering that linksthem up into a chain of dependences. Take, for instance, Suzy'srock in mid-trajectory. Certainly, this event depends onSuzy's initial throw, but the problem is that the shattering ofthe bottle does not depend on it, because even without it the bottlewould still have shattered because of Billy's throw. To be sure, the bottle shattering that would have occurred withoutSuzy's throw would be different from the bottle shattering thatactually occurred with Suzy's throw. For a start, it would haveoccurred later. This observation suggests that one solution to theproblem of late preemption might be to insist that the events involvedshould be construed as fragile events. Accordingly, it will be truerather than false that if Suzy had not thrown her rock, then theactual bottle shattering, taken as a fragile event with an essentialtime and manner of occurrence, would not have occurred. Lewis himselfdoes not endorse this response on the grounds that a uniform policy ofconstruing events as fragile would go against our usual practices, andwould generate many spurious causal dependences. For example, supposethat a poison kills its victim more slowly and painfully when taken ona full stomach. Then, the victim's eating dinner before he drinks thepoison would count as a cause of his death since the time and mannerof the death depend on the eating of the dinner. (For discussion ofthe limitations of this response see Lewis 1986c, 2000.) When we turn from preemption examples involving deterministiccausation to those involving chancy causation, we see that theproblems for Lewis's theory multiply. One particularly recalcitrantproblem is described in Menzies 1989. (See also Woodward 1990.)Suppose that two systems can produce the same effect, perhaps at thesame time and in the same manner. (It does not matter whether this isan example of early or late preemption.) However, one system is muchmore reliable than the other. The reliable system starts and, left toitself, will very probably produce the effect. But you do not leave itto itself. You throw a switch that shuts down the reliable system andturns on the unreliable one. As luck would have it, the unreliablesystem works and brings about the effect. This kind of examplepresents a problem for the probabilistic generalisation of thecounterfactual theory because the preempting actual cause decreasesthe chance of the effect while the preempted potential cause increasesits chance. In addition to the problem of explaining how thepreempting cause qualifies as a cause when the effect does notcausally depend on it, the probabilistic counterfactual theory facesthe problem of explaining how the preempted cause is not really acause when the effect does causally depend on it.(Examples of thiskind have been the subject of extensive discussion in the context ofboth counterfactual and probabilistic theories of causation. Fordiscussions about how best to deal with them within theories admittingof indeterminism,see Barker 2004; Beebee 2004; Dowe 2000, 2004;Hitchcock 2004; Kvart 2004; Noordhof 1999, 2004; Ramachandran 1997,2004.)4. Later DevelopmentsIn this section we shall consider some recent developments of thecounterfactual approach to causation, which have been motivated by thedesire to overcome the deficiencies in Lewis's 1973 theory, especiallywith respect to preemption.4.1 Lewis's 2000 Theory4.2 Causation as Intrinsic Relation4.3 The Structural Equations Framework4.1 Lewis's 2000 TheoryIn an attempt to deal with the various problems facing his 1973theory, Lewis developed a new version of the counterfactual theory,which he first presented in his Whitehead Lectures at HarvardUniversity in March 1999. (A shortened version of the lecturesappeared as his (2000). The full lectures are published as his(2004a).) Counterfactuals play a central role in the new theory, as in theold. But the counterfactuals it employs do not simply statedependences of whether one event occurs on whetheranother event occurs. The counterfactuals state dependences ofwhether, when, and how one event occurs onwhether, when, and how another event occurs. Akey idea in the formulation of these counterfactuals is that of analteration of an event. This is an actualised or unactualisedevent that occurs at a slightly different time or in a slightlydifferent manner from the given event. An alteration is, bydefinition, a very fragile event that could not occur at a differenttime, or in a different manner without being a different event. Lewisintends the terminology to be neutral on the issue of whether analteration of an event is a version of the same event or a numericallydifferent event. The central notion of the new theory is that of influence.(7)Where c and e are distinct events, cinfluences e if and only if there is a substantialrange of c1, c2, … of different not-too-distantalterations of c (including the actual alteration ofc) and there is a range of e1, e2, … ofalterations of e, at least some of which differ, such thatif c1 had occurred, e1 would have occurred, and ifc2 had occurred, e2 would have occurred, and so on. Where one event influences another, there is a pattern ofcounterfactual dependence of whether, when, and how upon whether,when, and how. As before, causation is defined as an ancestralrelation.(8) c causes e if and only if there is achain of stepwise influence from c to e. One of the points Lewis advances in favour of this new theory isthat it handles cases of late as well as early pre-emption. (Thetheory is restricted to deterministic causation and so does notaddress the example of probabilistic preemption described in section3.4.) Reconsider, for instance, the example of late preemptioninvolving Billy and Suzy throwing rocks at a bottle. The theory issupposed to explain why Suzy's throw, and not Billy'sthrow, is the cause of the shattering of the bottle. If we take analteration in which Suzy's throw is slightlydifferent (the rock is lighter, or she throwssooner), while holding fixed Billy's throw, we find thatthe shattering is different too. But if we make similar alterationsto Billy's throw while holding Suzy's throw fixed, we findthat the shattering is unchanged. Another point in favour of the new theory is that it handles a typeof preemption Lewis that have come to be calledtrumping. (Trumping was first described by Jonathan Schaffer:see his (2000).) Lewis gives an example involving a major and asergeant who are shouting orders at the soldiers. The major andsergeant simultaneously shout “Advance”; the soldiers hear them bothand advance. Since the soldiers obey the superior officer, theyadvance because the major orders them to, not because the sergeantdoes. So the major's command preempts or trumps the sergeant's. Whereother theories have difficulty with trumping cases, Lewis's argues hisnew theory handles them with ease. Altering the major's command whileholding fixed the sergeant's, the soldier's response would becorrespondingly altered. In contrast, altering the sergeant's command,while holding fixed the major's, would make no difference at all. There is, however, some reason for scepticism about whether the newtheory handles the examples of late preemption and trumpingcompletely satisfactorily. In the example of late preemption,Billy's throw has some degree of influence on the shattering ofthe bottle. For if Billy had thrown his rock earlier (so that itpreceded Suzy's throw) and in a different manner, the bottlewould have shattered earlier and in a different manner. Likewise, thesergeant's command has some degree of influence on thesoldiers' advance in that if the sergeant had shouted earlierthan the major with a different command, the soldiers would haveobeyed his order. In response to these points, Lewis must say thatthese alterations of the events are too-distant to be consideredrelevant. But some metric of distance in alterations is required,since it seems that similar alterations of Suzy's throw and themajor's command are relevant to their having causal influence. It has also been argued that the new theory generates a great number ofspurious instances of causation. (For discussion see Collins2000; Kvart 2001.) The theory implies thatany event that influences another event to a certain degree counts asone of its causes. But commonsense is more discriminating aboutcauses. To take an example of Jonathan Bennett (1987): rain inDecember delays a forest fire; if there had been no December rain,the forest would have caught fire in January rather than when it actuallydid in February. The rain influences the fire with respect to itstiming, location, rapidity, and so forth. But commonsense deniesthat the rain was a cause of the fire, though it allows thatit is a cause of the delay in the fire. Similarly, in theexample of the poison victim discussed above, the victim'singesting poison on a full stomach influences the time and manner ofhis death (making it a slow and painful death), but commonsenserefuses to countenance his eating dinner as a cause of his death,though it may countenance it as a cause of its being a slow andpainful death. Pace Lewis, commonsense does not take anythingthat affects the time and manner of an event to be a cause of theevent simpliciter.4.2 Causation as Intrinsic RelationOne way of treating preemption that has been recently discusseddeparts from a purely counterfactual analysis of causation. It hasbeen argued that preemption examples highlight the intuitive ideathat causation is an intrinsic relation between events, which is tosay it is a local relation depending on the intrinsic properties ofthe events and what goes on between them, and nothing else. Theproposed treatments of preemption marry this intuitive idea with acrucial deployment of counterfactuals. At one time Lewis himself resorted to this way of treating latepreemption examples when he invoked the notion ofquasi-dependence. (See his (1986c).) To explain this notionconsider a case that resembles the case of Billy and Suzy throwingrocks at a bottle. Suzy throws a rock and shatters the bottle inexactly the same way in which she does in the original case. But inthis case Billy and his rock are entirely absent. Lewis argued thatsince the process in the original case and the process in thecomparison case are intrinsically alike (and also obey the samelaws), both or neither must be causal. However, the comparisonprocess is surely a causal process since, thanks to Billy'sabsence, it exhibits a causal dependence. Accordingly, the process inthe original case must be a causal process too, even though it doesnot exhibit a causal dependence. In such examples Lewis has said that theactual process that does not exhibit causal dependence is,nonetheless, causal by courtesy: it exhibits quasi-dependencein virtue of its intrinsic resemblance to the causal process in thecomparison case. A related idea is pursued in Menzies (1996; 1999). Menzies arguesthat there is an element in our concept of causation that resistscapture in purely counterfactual terms. This element consists in theidea that causation is a structural relation that underlies andsupports causal dependences. This idea can be captured by treatingthe concept of causation as the concept of a theoreticalentity. Applying a standard treatment of theoretical concepts, heargues that causation should be defined as the unique occupant of acertain characteristic role given by the platitudes of the folktheory of causation. One platitude is that causation is an intrinsicrelation between events. Another platitude is that it is typically,but not invariably, accompanied by causal dependence. Accordingly,causation is defined in the following way:(9) c causes e if only if the intrinsic relationthat typically accompanies causal dependence holds between cand e. On this account, causation is not constituted by causaldependence. It is, in fact, a distinct relation for which causaldependence is, at best, a defeasible marker. The relation may beidentified a posteriori with some physically specificablerelation such as energy-momentum transfer. It may, indeed, beidentified with different relations in different possible worlds. This definition is supposed to explain commonsense intuitions aboutpreemption examples. For example, Suzy's throw, and notBilly's throw, caused the shattering of the bottle, because theintrinsic relation that typically accompanies causal dependenceconnects Suzy's throw, but not Billy's throw, with theshattering of the bottle. Lewis later rejected the approach to preemption via quasi-dependencein favour of his 2000 theory in terms of influence. In Lewis 2004a and2004b, he claims that theories of causation as an intrinsic relationdo not do justice to the full range of our intuitions aboutcausation. (For related points see Hall 2002, 2004.) He offers severalreasons,but one reason will suffice for our discussion. The intuitionthat causation is an intrinsic matter does not apply to cases ofdouble prevention. Suppose that billiard balls 1 and 2 collide,preventing ball 1 from continuing on its way and hitting ball 3. Ifthe collision of balls 1 and 3 had occurred, ball 3 would not havelater collided with ball 4. So, we have double prevention: thecollision of balls 1 and 2 prevented the collision of balls 1 and 3,which would have prevented the later collision of balls 3 and 4. Hereit seems reasonable to say that the collision of balls 1 and 2 was acause of the later collision of balls 3 and 4. Lewis observes that thecausation in such cases of double prevention is partly an extrinsicmatter. If there had been some other obstruction that would havestopped ball 1 from hitting ball 3, the collision of 3 and 4 would nothave depended on the collision of 1 and 2. Moreover, he notes thatmuch of the spatiotemporal region between the collision of balls 1 and2 and the collision of balls 3 and 4 is simply empty so that there isno chain of events to serve as a connecting process between cause andeffect. The intuition that causation is an intrinsic relation does notapply in this case. More generally, he argues that theories ofcausation as an intrinsic relation are overhasty generalisations ofone specific kind of causation, and they fail to do justice to ourintuitions about causation involving absences (as causes, effects orintermediaries).4.3 The Structural Equations Framework A number of contemporary philosophers (Hitchcock 2001, 2007; Woodward2003; Woodward and Hitchcock 2003) have explored an alternativecounterfactual approach to causation that employs the structuralequations framework. This framework, which has been used in the socialsciences and biomedical sciences since the 1930s and 1940s, receivedits state-of-the-art formulation in Judea Pearl's landmark 2000book. Hitchcock and Woodward acknowledge their debt to Pearl's workand to the related work on causal Bayes nets by Peter Spirtes, ClarkGlymour, and Richard Scheines (1993). However, while Pearl andSpirtes, Glymour and Scheines focus on issues to do with causaldiscovery and inference, Woodward and Hitchcock focus on issues of themeaning of causal claims. For this reason, their formulations of thestructural equations framework are better suited to purposes of thisdiscussion. The exposition of this section follows that of Hitchcock2001, in particular. While philosophical work using this framework hasonly just begun, it would seem that this framework looks likely torival Lewis's framework in terms of its theoretical richness andfruitfulness. The structural equations framework describes the causal structure ofa system in terms of a causal model of the system, which is identifiedas an ordered pair <V, E>, where V is a set ofvariables and E a set of structural equations statingdeterministic relations among the variables. (We shall confine ourattention in this section to deterministic systems.) The variables inV describe the different possible states of the system inquestion. While they can take any number of values, in the simpleexamples to be considered here the variables are typically binaryvariables that take the value 1 if some event occurs and the value 0if the event does not occur. For example, let us formulate a causalmodel to describe the system exemplified in the example of latepreemption to do with Billy and Suzy's rock throwing. We mightdescribe the system using the following set of variables:BT = 1 if Billy throws a rock, 0 otherwise;ST = 1 if Suzy throws a rock, 0 otherwise;BH = 1 if Billy's rock hits the bottle, 0 otherwise;ST = 1 if Suzy's rock hits the bottle, 0 otherwise;BS = 1 if the bottle shatters, 0 otherwise. Here the variables are binary. But a different model might have usedmany-valued variables to represent the different ways in which Billyand Suzy threw their rocks, their rocks hit the bottle, or the bottleshattered. The structural equations in a model describe the dynamical evolutionof the system being modelled. There is a structural equation for eachvariable. The form taken by a structural equation for a variabledepends on which kind of variable it is. The structural equation foran exogenous variable (the values of which are determined byfactors outside of the model) takes the form of Y = y,which simply states the actual value of the variable. The structuralequation for an endogenous variable (the values of which aredetermined by factors within the model) states how the value of thevariable is determined by the values of the other variables. It takesthe form:Y = f(X1,…, Xn) What does this structural equation mean? There are in fact competinginterpretations. The interpretation favoured by Woodward and Hitchcockis that the equation for an endogenous variable encodes a set ofcounterfactuals of the following form: If it were the case that X1 = x1,X2 = x2,…, Xn= xn, then it would be the case that Y =f(x1,…,xn).As this form of counterfactual suggests, the structural equations areto be read from right to left: the antecedent of the counterfactualstates possible values of the variables X1 throughto Xn and the consequent states the correspondingvalue of the endogenous variable Y. There is a counterfactualof this kind for every combination of possible values of the variablesX1 through to Xn. It is importantto note that a structural equation of this kind is not, strictlyspeaking, an identity that is equivalent to f(X1,…,Xn) = Y: there is a right-to-left asymmetrybuilt into the equation. Another important feature of the structuralequations for endogenous variables is that they must be complete inthe sense that the equation for a variable Y must express thevalue of Y as a function of all and only the variablesXi on which it counterfactually depends given thevalues of the other variables. A crucial question for those interestedin the semantic and metaphysical foundations of the structuralequations framework is the status of the counterfactuals encoded bythe structural equations. Are they semantically and metaphysicallyprimitive so that the structural equations are simply a summary of themore basic counterfactuals? Or are the structural equations themselvesto be taken as the conceptual and metaphysical primitives, with thecounterfactuals having a secondary, derivative status? So far there isno consensus on the best way to answer these questions. As an illustration, consider the set of structural equations thatmight be used to model the late preemption example of Billy andSuzy. Given the variables listed above, the structural equations mightbe stated as follows:ST = 1;BT = 1;SH = ST;BH = BT & ~SH;BS = SH v BT. In these equations logical symbols are used to represent mathematicalfunctions on binary variables: ~X = 1 − X;X v Y = max{X, Y}; X & Y =min{X, Y}. The first two equations simply state theactual values of the exogenous variables ST and BT. Thethird equation encodes two counterfactuals, one for each possiblevalue of ST. It states that if Suzy threw a rock, her rock hitthe bottle; and if she didn't throw a rock, her rock didn't hit thebottle. The fourth equation encodes four counterfactuals, one for eachpossible combination of values for BT and ~SH. It statesthat if Billy threw a rock and Suzy's rock didn't hit the bottle,Billy's rock hit the bottle; but didn't do so if one or more of theseconditions was not met. The fifth equation encodes fourcounterfactuals, one for each possible combination of values forSH and BH. It states that if one or other (or possiblyboth) of Suzy's rock or Billy's rock hit the bottle, the bottleshattered; but if neither rock hit the bottle, the bottle didn'tshatter. The structural equations above can be represented in terms of adirected graph. The variables in the set V are representedas nodes in the graph. An arrow directed from one node X toanother Y represents the fact that the variable Xappears on the right-hand side of the structural equation forY. In this case, X is said to be a parent ofY. Exogenous variables are represented by nodes that have noarrows directed towards them. A directed path from X toY in a graph is a sequence of arrows that connect X withY. The directed graph of the model described above of Billy andSuzy example is depicted in Figure 1 below:  Figure 1 The arrows in this figure tell us that the bottle's shattering is afunction of Suzy's rock hitting the bottle and Billy's rock hittingthe bottle; that Billy's rock hitting the bottle is a function ofBilly's throwing a rock and Suzy's rock hitting the bottle; and thatSuzy's rock hitting the bottle is a function of her throwing therock. (The existence of an arrow from one variable to another does notalways signify a stimulatory connection. For example, the arrowdirected from SH to BH is inhibitory.) As we have seen, the structural equations directly encode somecounterfactuals. However, some counterfactuals that are not directlyencoded can be derived from them. Consider, for example, thecounterfactual “If Suzy's rock had not hit the bottle, it would stillhave shattered”. As a matter of fact, Suzy's rock did hit thebottle. But we can determine what would have happened if it hadn'tdone so, by replacing the structural equation for the endogenousvariable SH with the equation SH = 0, keeping all theother equations unchanged. So, instead of having its value determinedin the ordinary way by the variable ST, the value of SHis set “miraculously”. Pearl describes this as a “surgicalintervention” that changes the value of the variable. In terms of itsgraphical representation,this amounts to wiping out the arrow from thevariable ST to the variable SH and treating SH asif it were an exogenous variable. After this operation, the value ofthe variable BS can be computed and shown to be equal to 1:given that Billy had thrown his rock, his rock would have hit thebottle and shattered it. So this particular counterfactual istrue. This procedure for evaluating counterfactuals has directaffinities with Lewis's non-backtracking interpretation ofcounterfactuals: the surgical intervention that sets the variableSH at its hypothetical value but keeps all other equationsunchanged is similar in its effects to Lewis's small miracle thatrealises the counterfactual antecedent but preserves the past. In general, to evaluate a counterfactual, say “If it were the casethat X1,…,Xn, then …”, one replacesthe original equation for each variable Xi with anew equation stipulating its hypothetical value,while keeping theother equations unchanged; then one computes the values for theremaining variables to see whether they make the consequent true. Thistechnique of replacing an equation with a hypothetical value set by a“surgical intervention” enables us to capture the notion ofcounterfactual dependence between variables:(10)A variable Y counterfactually depends on a variable X in a model if and only if it is actually the case that X = x and Y = y and there exist values x' ≠ x and y' ≠ y such that replacing the equation for X with X = x' yields Y = y'. How does the structural equations framework deal with examples oflate pre-emption that pose such problems for Lewis's counterfactualtheory? Can this framework deliver the intuitively correct verdicts inthe example about Suzy and Billy? Halpern and Pearl (2001,2005),Hitchcock (2001),and Woodward (2003a) all give roughly the sametreatment of examples of late preemption. The key to their treatmentis the employment of a certain procedure for testing the existence ofa causal relation. The procedure is to look for an intrinsic processconnecting the putative cause and effect; suppress the influence oftheir noninstrinsic suroundings by “freezing” those surroundings asthey actually are; and then subject the putative cause to acounterfactual test. So, for example, to test whether the variableSuzy's throwing a rock caused the bottle to shatter, we shouldconsider the examine the process running from ST throughSH to BS; hold fix at its actual value the variableBH which is extrinsic to this process; and then wiggle thevariable ST to see if it changes the value of BS. Thelast steps involve evaluating the counterfactual “If Suzy hadn'tthrown a rock and Billy's rock hadn't hit the bottle, the bottle wouldnot have shattered”. It is easy to see that this counterfactual istrue. In contrast, when we carry out a similar procedure to testwhether Billy's throwing a rock caused the bottle to shatter,we arerequired to consider the counterfactual “If Billy hadn't thrown hisrock and Suzy's rock had hit the bottle, the bottle would notshattered”. This counterfactual is false. It is the difference in thetruth-value of these two counterfactuals that explains the fact thatit was Suzy's rock throwing, and not Billy's, that caused the bottleto shatter. Hitchcock (2001) presents a useful regimentation of thisreasoning. He defines a route between two variables Xand Z in the set V to be an ordered sequence ofvariables <X, Y1,…,Yn, Z> such each variable in the sequenceis in V and is a parent of its successor in the sequence. Avariable Y is intermediate between X and Zif and only if it belongs to some route between X andZ. Then he introduces the new concept of an active causalroute:(11)The route <X, Y1,…,Yn, Z> is active in the causal model<V, E> if and only if Z depends counterfactuallyon X within the new system of equations E' constructedfrom E as follows: for all Y in V, if Y isintermediate between X and Z but does not belong to theroute <X, Y1,…, Yn,Z>, then replace the equation for Y with a new equationthat sets Y equal to its actual value in E. (If thereare no intermediate variables that do not belong to this route, thenE' is just E.) This definition generalises the informal idea sketched in the exampleof Suzy and Billy. There is an active causal route going from Suzy'sthrowing her rock through her rock hitting the bottle to the bottleshattering: when we hold fixed Billy's rock not hitting the bottle,which is the actual value of the only intermediate variable BHthat is not on this route, we see that the bottle's shatteringcounterfactually depends on Suzy's throwing her rock. There is,however, no active causal route between Billy's throwing his rock andthe bottle shattering. In terms of the notion of an active causal route, Hitchcock definesactual or token causation in the following terms:(12)If c and e are distinct actual events and X and Z are binary variables whose values represent the occurrence and non-occurrence of these events, then c is a cause of e if and only if there is an active causal route from X to Z in an appropriate causal model <V, E>. A crucial notion in this definition is that of “an appropriate”model. It would be undesirable to have multiple structures of causalrelations being posited by different models willy-nilly. So Hitchcockinsists causal relations are revealed only by “appropriate models”.He mentions a number of criteria for appraising whether a model isappropriate, the most important one being that the structuralequations posited by the model must not imply any falsecounterfactual. In order to deal with examples of symmetricoverdetermination, Hitchcock (2001) defines a notion of a weaklyactive route, the essential idea being that there is a weaklyactive route between X and Y just when Ycounterfactually depends on X under the freezing of somepossible, not necessarily actual, values of the variables that are noton the route from X to Y. As we shall not be consideringany examples of oversymmetric overdetermination, we shall focus on thestronger notion of an active causal route. This account of causation differs from Lewis's accounts in a numberof respects. One difference is that the account does not appeal to thetransitivity of causation to deal with preemption examples, incontrast to Lewis's accounts, both early and late. Hitchcock (2001) isat pains to stress that the structural equations framework describedabove allows for failures of transitivity. Another difference betweenthe accounts is that the structural equations account appeals tospecial counterfactuals with complex antecedents in order to handlepreemption examples. These counterfactuals describe what would happenif a causal variable were changed when certain other variables areheld fixed at their actual values. (Hitchcock calls these “explicitlynonforetracking counterfactuals”.) Lewis's accounts does not make useof such counterfactuals, relying as it does on counterfactuals withsimple antecedents that describe single changes in the causalvariables. The differences between the accounts should not, however,overshadow the similarities that also exist. Both accounts makecentral use of non-backtracking counterfactuals and they interpretthese counterfactuals in roughly the same fashion. Setting asidecomplications to do with backwards causation, Lewis's account and thestructural equations account have us evaluate a non-backtrackingcounterfactual in much the same way: we are to hold fixed the pasthistory of the system, imagine that the antecedent is realised“miraculously” by a surgical intervention from outside the system, andthen consider how the new state of the system would evolve inconformity with the structural equations or laws of the system withoutany further interventions. How plausible is this new counterfactual approach to causation? It istoo early to say with any confidence, as the approach is still beingdeveloped and it has not been subjected to sustained, rigoroustesting. Nonetheless, some early problems have emerged. (See Hall2007; Hitchcock 2007; and Menzies 2004b.) Consider, for instance, thefollowing example, which is a variant of one described by Hitchcock(2007). An assassin puts poison in the king's coffee. The bodyguardresponds by pouring an antidote in the king's coffee. If the bodyguardhad not poured the antidote in the coffee, the king would havedied. On the other hand, the antidote is fatal when taken by itself;and if the poison had not been poured in first, it would have killedthe king. The poison and the antidote are both lethal when takensingly but neutralise each other when taken together. In fact, theking drinks the coffee and survives.Suppose we model this scenario using the following variables: A = 1 if the assassin pours poison into the king's coffee, 0 otherwise;G = 1 if the bodyguard responds by pouring antidote into the coffee, 0 otherwise;S = 1 if the king survives, 0 otherwise.And also suppose that we employ these structural equations:A = 1;G = A;S = (A & G) v (~A & ~G).The directed graph for this model is depicted in Figure 2.  Figure 2 Testing for active causal processes, we can see that the process thatgoes directly from the assassin's pouring the poison in the coffee tothe king's survival is active. Holding fixed the fact that thebodyguard poured the lethal antidote into the coffee, we note that theking would not have survived if the assassin had not put the poison inthe coffee first. So the theory licenses the verdict that theassassin's pouring in the poison caused the king to survive. However,many regard this as a mistaken causal verdict: putting poison in theking's coffee is exactly the kind of thing that is likely to kill theking. It might be argued that the causal verdict is justified in viewof the fact that the assassin's action caused the bodyguard's action,which in turn caused the king's survival. But this appeal to thetransitivity of causation is not open to the defenders of this theory,who deny the validity of transitivity. One counterexample by itself is not enough to disprove the wholestructural equations framework. Strictly speaking, it only casts doubton the theory of causation that defines causation in terms of thepresence of an active causal route. Perhaps there are alternativedefinitions within the structural equations framework that farebetter. Indeed, a number of philosophers have explored the possibilityof framing a better theory by appealing to a distinction between whatHitchcock has called “default” and “deviant” values of variables.(SeeHitchcock 2007.) The default value of some variable represents anormal or to-be-expected state of the system, whereas a deviant valuerepresents an abnormal or unusual state of the system. The correlativenotion of the default course of evolution for a system can becharacterised as a temporally-ordered sequence of values that thevariables in a model take when the default values of the exogenousvariables are plugged into the structural equations of themodel. Thus, if we set the value of the exogenous variable A inthe example above at its default value 0 instead of its actual value1, we can see that the scenario described above will evolve in thefollowing way: the assassin doesn't put the poison in the coffee, thebodyguard doesn't put the antidote into the coffee, and the kingsurvives. Now if we evaluate counterfactual dependences withcounterfactuals centred on the default course of evolution rather thanthe actual course of evolution, we can see that the bodyguard's actioncounterfactually depends on the assassin's action and the king'ssurvival depends on the bodyguard's action, but the king's survivaldoesn't depend on the assassin's action. If counterfactual dependencescentred on the default course of evolution are taken to indicatecausal relations, these counterfactual dependences accurately reflectour intuitive causal judgements. (For further discussion of this idea,see Menzies 2004a, 2004b, 2007.) 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