Many different accounts of causation, of a Humean reductionist sort, have been advanced, but four types are especially important.  Of these, three involve analytical reductionism.  First, there are approaches which start out from the general notion of a law of nature, then define the ideas of necessary, and sufficient, nomological conditions, and, finally, employ the latter concepts to explain what it is for one state of affairs to cause another.  Secondly, there are approaches that employ subjunctive conditionals, either in an attempt to give a purely counterfactual analysis of causation (David Lewis, 1973 and 1979), or as a supplement to other notions, such as that of agency (Georg von Wright, 1971).   Thirdly, there are approaches that employ the idea of probability, either to formulate a purely probabilistic analysis (Hans Reichenbach, 1956;  I. J. Good, 1961 and 1962;  Patrick Suppes, 1970; Ellery Eells, 1991; D. H. Mellor, 1995) - where the central idea is that a cause must, in some way, make its effect more likely -- or as a supplement to other ideas, such as that of a continuous process (Wesley Salmon, 1984).  Finally, a fourth approach involves the idea of offering, not an analytic reduction of causation, but a contingent identification of causation, as it is in this world, with a relation whose only constituents are non-causal properties and relations.  One idea, for example, is that causal processes can be identified with continuous processes in which relevant quantities are conserved (Wesley Salmon, 1997 and 1998;  Phil Dowe, 2000a and 2000b).

 5.1  Causes and Nomological Conditions

         One very familiar approach to causation involves attempting to analyze causation in terms of nomological concepts.  Given the idea of a law of nature, one can define what it is for a state of affairs to be a nomologically necessary condition of some other state of affairs, or a nomologically sufficient condition of another states of affairs.  Similarly, one can define what it is for a states of affairs to be nomologically necessary in the circumstances, or nomologically sufficient in the circumstances, for another state of affairs.  The proposal is then that what it is for one state of affairs to cause another can be analyzed in terms of these nomological concepts.

         According to one version, a cause is a condition that is necessary in the circumstances for its effect, where to say that event c is necessary in the circumstances for event e is roughly to say that there is some law, l, and some circumstance, s, such that the non-occurrence of c, in circumstance s, together with law l, logically entails the non-occurrence of e.  (Ignoring temporal constraints upon the relation between cause and effect, this answer is essentially that advanced by Ernest Nagel (1961 pp. 559-60).   It is also considered seriously, but rejected, by Michael Scriven (1966, particularly section 8, pp. 258-62), while a very similar view is defended by Raymond Martin (1972, pp. 205-11).)

         Alternatively it may be held instead that a cause is a condition that is sufficient in the circumstances for its effect, where to say that event c is sufficient in the circumstances for event e is to say that there is some law, l, and some circumstance, s, such that the occurrence of c, in circumstance s, together with law l, logically entails the occurrence of e.  (If, once again, we ignore the addition of temporal constraints, this answer is essentially equivalent, for example, to views advanced by John Stuart Mill (1874, Book 3, ch. 5), R. B. Braithwaite (1953, pp. 315-8), H. L. A. Hart and A. M. Honoré (1959, pp. 106-7), C. G. Hempel (1965, p. 349),  and Karl Popper (1972, p. 91).)

         Another possibility is that for one event to cause another is for its occurrence to be both necessary and sufficient in the circumstances for the occurrence of the other event -- a view that was seriously entertained, but ultimately rejected, by Richard Taylor (1966, ch. 3).

         These accounts, however, are open to a number of very serious objections.  First, it seems very plausible, especially in view of quantum physics, that probabilistic causal laws are logically possible, and while such laws do not preclude there being nomologically necessary conditions for a given type of event, they do entail that there are no nomologically sufficient conditions.  So if probabilistic causal laws are possible, all of the above accounts, except for the first, are ruled out.

         Secondly, all of the above fall prey to the underdetermination problem, set out in the preceding section.

         Thirdly, it would certainly seem that there could be laws that are not causal -- such as, for example, Newton's Third Law of Motion.  But given that law, all of the above analyses have the unacceptable consequence that A's exerting a certain force on B at a given time causes B to exert an equal and opposite force on A at that very same time.

         One way of attempting to escape this objection would be by reformulating the account in terms of basic laws, and then arguing that all non-causal laws must be derivable from causal laws.  But it is not at all easy to see how one might establish the latter thesis.

         Finally, and most seriously of all, no account of causation in terms of nomological relations alone can provide any account of the direction of causation.  Thus, if our world were a Newtonian one, where the basic laws are time-symmetric, the total state of the universe in 1950 would have been both necessary and sufficient not only for the total state in 2050, but also for the total state in 1850.  It would therefore follow, on any of the above accounts, that events in 1950 had caused events in 1850.

         The only way to escape this problem within the context of this general approach is by adding the requirement that one event can be the cause of another event only if the one is temporally prior to the other.  To make this part of the definition of a cause seems, however, unsatisfactory.  For while it may be true that a cause necessarily precedes its effect, if this is true, it should be a deep analytical result, not an immediate consequence of the analysis of causation --` given that readers and writers of science fiction have certainly thought that they could imagine scenarios involving backward causation.

 5.2  Counterfactual Approaches

         A second important reductionist approach attempts to analyze causation in terms of counterfactuals.  Such approaches come in different forms, and can be arrived at via different routes.  One way of arriving at a counterfactual account is by analyzing causation in terms of necessary and/or sufficient conditions, but then interpreting the latter, not in terms of  nomologically necessary and nomologically sufficient conditions, but in terms of subjunctive conditionals.  Thus, one can say that c is necessary in the circumstances for e if, and only if, had c not occurred, e would not have occurred, and that c is sufficient in the circumstances for e if, and only if, had e not occurred, c would not have occurred.

         John Mackie took this tack in developing a more sophisticated analysis of causation in terms of necessary and sufficient conditions.  Thus, after defining an INUS condition of an event as an insufficient but necessary part of a condition which is itself unnecessary but exclusively sufficient for the event, and then arguing that c's being a cause of e can then be analyzed as c's being at least an INUS condition of e, Mackie asked how necessary and sufficient conditions should be understood.  For general causal statements, Mackie favored a nomological account, but for singular causal statements, he argued for an analysis in terms of subjunctive conditionals (1973, p. 48).

         The most fully worked out counterfactual approach, however, is that of David Lewis (1973).   His original, basic strategy involved analyzing causation in terms of a narrower notion of causal dependence, and then analyzing causal dependence counterfactually:  (1) an event c causes an event e if and only if there is a chain of causally dependent events linking e with c;  (2) an event g is causally dependent upon an event f if and only if, had f not occurred, g would not have occurred.

         Causes, so construed, need not be necessary for their effects, since counterfactual dependence, and hence causal dependence, are not necessarily transitive.  Nevertheless, Lewis's approach is very closely related to necessary condition analyses of causation, since the more basic relation of causal dependence is a matter of one event's being counterfactually necessary in the circumstances for another event.

 5.2.1  Some Important Objections to David Lewis’s Counterfactual Analysis of Causation

         How satisfactory are analyses of causation in terms of counterfactuals?  One objection to Lewis's approach is that it is formulated in terms of events, and it then becomes a delicate matter to set out an account of the individuation of events that will not generate unwelcome consequences concerning causal relations (Lewis, 1986b, and Bennett, 1998).   A much better approach, it would seem, would be to view the basic causal relata as states of affairs - or as events in Jaegwon Kim’s sense (1971, 1973a) - and thus to regard the basic singular causal statements as those that explicitly specify the causally relevant factors, and that do not incorporate causally extraneous information.  For not only does this seem metaphysically more perspicuous, it also enables one to avoid getting one's account of causation entangled in the problem of the individuation of events.

         Of course, we certainly make causal statements that provide no information at all about what properties and/or relations enter into the causally relevant states of affairs - such as "Mary's remark caused an interesting occurrence."   But it would seem to be a relatively straightforward matter to analyze such event-statements in terms of metaphysically more basic statements concerning causal relations between states of affairs.

         A second objection - originally advanced by Jaegwon Kim in his article, "Causes and Counterfactuals” (1973b) - focuses upon the fact that there are a number of counterfactuals that have nothing to do with causation.  If, for example, John and Mary are married at time t, it is true that if John had not existed at time t, then Mary would not have been married at time t.  But John's existing at time t is not a cause of the simultaneous state of affairs that is Mary's being married at time t.

         How might this objection be handled?  There has been relatively little discussion of this problem, but it would seem that one will have to draw a line between counterfactuals whose truth depends upon laws of nature, and those whose truth does not so depend.  Exactly how this is to be done, given the type of approach to counterfactuals that must be employed here, is not entirely clear.

         A third objection is that some counterfactuals are based upon non-causal laws.  Thus, for example, counterfactuals such as "If A had not exerted force F upon object B, then B would not have exerted a force G upon A" will be true in Newtonian worlds by virtue of Newton's Third Law of Motion.  On Lewis's counterfactual analysis, it follows that A's exerting force F on B causes B's exerting force G on A, and vice versa - which is surely wrong.

          A fourth objection involves overdetermination, or redundant causation, where two events, C and D, are followed by an event E, and where each of C and D would have been causally sufficient, on its own to produce E.  If it is true that C causes E and that D causes E, then one has a counterexample to Lewis's counterfactual analysis.  Lewis contends that we are uncertain what to say here.  Do C and D each cause E, or do they jointly cause E?  But is Lewis right about this?  If, for example, Lewis (1973, p. 73) were right in holding that "a contingent generalization is a law of nature if and only if it appears as a theorem (or axiom) in each of the deductive systems that achieves a best combination of simplicity and strength," then it would seem that it would have to be the case that C causes E and D causes E, since more complicated generalizations are needed if one is to say instead that it is only the combination of C together with D that causes E.  Similarly, if simplicity is, instead, epistemologically relevant, will not the conclusion be the same?  So overdetermination certainly seems to be a problem.

         A fifth objection involves cases of preemption - where, once again, one has causation without causal dependence.  Until recently, the discussion of preemption had focused on cases where one causal process preempts another by blocking the occurrence of some state of affairs in the other process, and a variety of closely related ways of attempting to handle this type of preemption have been advanced, involving such notions as fragility of events, quasi-dependence, continuous processes, minimal-counterfactual sufficiency, and minimal-dependence sets (Lewis, 1986c; Menzies, 1989; McDermott, 1995; Ramachandran, 1997).  But none of these approaches can handle the case of trumping preemption, advanced by Jonathan Schaffer, in his article "Trumping Preemption" (2000).

         David Lewis's own reaction to the problem posed by trumping preemption has been to replace his previous counterfactual accounts by a new, 'causation as influence', account:

 Where C and E are distinct actual events, let us say that C influences E if and only if there is a substantial range C1, C2, . . . of different not-too-distant alterations of C (including the actual alteration of C) and there is a range E1, E2 . . . of alterations of E, at least some of which differ, such that if C1 had occurred, E1 would have occurred, and if C2 had occurred, E2 would have occurred, and so on.  (2000, p. 190)

         But this account does not really provide an answer to the trumping preemption objection.  For suppose that, contrary to what is required for Lewis's idea of causation as influence to be applicable, there is not a substantial range C1, C2 . . . of different not-too-distant alterations of C: there is only C, or its absence.  Suppose further that there is a substantial range of alterations of D - D1, D2, and so on - and where, in the absence of C, D will give rise to E, D1 to E1, D2 to E2, and so, and where E, E1, E2, etc. are all distinct.  Suppose, finally, that if C accompanies any of D, D1, D2, etc., then it is always E that comes about.  Then surely the simplest hypothesis will involve laws according to which C preempts all of D, D1, D2, etc.  Consequently, there are cases of trumping preemption that Lewis's revised account cannot handle.

 5.2.2  The Fundamental Objection to Counterfactual Analyses of Causation

         The preceding objections are all, I think, important ones, and I am inclined to think that at least some of them are likely to constitute decisive objections to the counterfactual approach to causation.  However, as the ongoing discussions of, for example, preemption show, to show that any of these objections provides a refutation of all counterfactual analyses of causation calls for considerable work.  My goal in this paper, accordingly, is to pursue a different line of attack, and one that can, I think, be shown to be decisive.
 
         If causation is to be analyzed counterfactually, one needs to show that there is a satisfactory account of counterfactuals that is compatible with such an analysis.  Many accounts of counterfactuals are clearly not available, since they incorporate causal concepts.  This is so, for example, of the analysis of counterfactuals advanced by Frank Jackson in his article, “A Causal Theory of Counterfactuals” (1977), and it is also the case for the somewhat complex account advanced by Igal Kvart in his book, A Theory of Counterfactuals (1986).  The question, then, is whether it is possible to provide a satisfactory analysis of counterfactuals without employing causal notions.  If it can be shown that no such account is available, then a counterfactual analysis of causation does not even get started.

 5.2.3  The Stalnaker-Lewis Approach to Counterfactuals

         There is, of course, an obvious candidate to play this role: a Stalnaker-Lewis-style analysis of counterfactuals.  After all, many philosophers today regard this approach as the standard account of the truth conditions of counterfactuals.  So why is there a problem?  The answer, as I shall try to show, is that there are decisive objections to a Stalnaker-Lewis approach to counterfactuals.

 5.2.3.1  The Nature of the Account

         This general approach to counterfactuals­which appeals to similarity relations between possible worlds­was first set out by Robert Stalnaker in his 1968 article, "A Theory of Conditionals,"  and then a modified, and in some ways more satisfactory version of it was advanced by David Lewis in his 1973 book, Counterfactuals.

         David Lewis's detailed exposition, in his book Counterfactuals, of this type of account of counterfactuals elicited some very important criticisms of the whole approach, including ones advanced in early reviews by Jonathan Bennett  and Kit Fine.   But Lewis, in "Counterfactual Dependence and Time's Arrow,"  and then in his "Postscripts to 'Counterfactual Dependence and Time's Arrow',"  attempted to show that, given, for example, the right sort of account of the factors that enter into judgments of similarity between possible worlds in the case of counterfactuals, one could escape the most crucial objections that had been advanced.

         One of my goals in this paper is to show that while the answer that Lewis offered to the most fundamental objection advanced by Jonathan Bennett and Kit Fine is successful in blocking the objection as they stated it, it fails when confronted with a reformulation of that objection.  But I shall also be arguing that the approach is exposed to other, very strong objections.

 5.2.3.1.1  Robert Stalnaker's Proposal in "A Theory of Conditionals"

         In his article, "A Theory of Conditionals," Robert Stalnaker offered the following, "selection-function" account of counterfactuals:

 " . . . our semantical apparatus includes a selection function, f, which takes a proposition and a possible world as arguments and a possible world as its value.  The s-function selects, for each antecedent A, a particular possible world in which A is true.  The assertion which the conditional makes, then, is that the consequent is true in the world selected.  A conditional is true in the actual world when its consequent is true in the selected world."

         But exactly what idea is the selection function supposed to be capturing?  Stalnaker's answer is that the informal truth conditions that he proposed earlier in the article "required that the world selected differ minimally from the actual world."  Thus, "the selection is based on an ordering of possible worlds with respect to their resemblance to the actual world."

         What reason did Stalnaker offer for embracing this sort of account?  The answer is that his acceptance does not seem to rest upon much beyond some criticisms that he advances against a particular type of alternative - namely, one that appeals to logical or causal connections between the antecedent and the consequent of a counterfactual.  But such criticism is very narrowly directed.  It does not, for example, give one any reason for rejecting causal approaches in general, since the criticism in question does not tell against approaches according to which "If p were the case, then q would be the case" can be true even though p does not itself have any causal or logical connection to q, because, for example,  there is some r that is compatible with p, and which is true, and which is causally related to q.

 5.2.3.1.2  David Lewis’s Account

         In his book, Counterfactuals, David Lewis offered the following, "ordering relation" account of counterfactuals:

 f ÅÆ y  is true at a world i (according to a system of spheres $) if and only if either

(1) no f-world belongs to any sphere S in $i, or
(2) some sphere S in $i does contain at least one f-world, and f … y holds at every world in S.

        What is the "system of spheres"?  The basic idea is that for any possible world, all other possible worlds can be placed on spheres that are centered on the world in question, with the size of a given sphere representing how close each world on the sphere is to the world that lies at the center of the given system of spheres.  Thus, all worlds on a given sphere are equally similar to the world at the center, and if one sphere is inside another, then the worlds on the inner sphere are more similar to the world at the center than are the worlds on the outer sphere.

         Accordingly, if one replaces the reference to a system of spheres by a direct reference to similarity, the account of the truth conditions of counterfactuals that Lewis is advancing is roughly as follows:

 The counterfactual "If p were the case, then q would be the case" is true in world W

 if and only if

 Either (1) there is no world at all where p is true, or else (2) some world W* in which both p and q are true is closer to W than any world in which p is true and q is false.

 5.2.3.2  A Crucial Objection to the Theory:  Jonathan Bennett and Kit Fine

         In his review of Lewis's book, Counterfactuals, Jonathan Bennett advanced a number of objections to a similarity-over-possible-worlds approach to counterfactuals.  But Bennett suggested that the "fatal defect" in the whole approach was that it either generated the wrong truth-values for certain counterfactuals, or else it involved unsound judgments of similarity.

         Bennett illustrated his point by a counterfactual concerning Oswald and the death of Kennedy:

 "If Oswald had not killed him, Kennedy would not have been killed."

         Suppose, Bennett says, that the Warrenite hypothesis that Oswald acted alone, etc., is true.  Then Lewis's approach will generate the wrong truth-value for the above counterfactual, since it is, Bennett claims, "incredible" that "some worlds in which no one kills Kennedy are more like the actual world than is any world in which Kennedy is killed by someone other than Oswald."

         Similarly, Kit Fine, in his review of Lewis's book, also argued that Lewis's approach generates the wrong truth-values for counterfactuals where the consequent could only be true if the world were radically different from the actual world, and Fine illustrated this point  by the following counterfactual concerning Nixon and the button:

 "If Nixon had pressed the button, there would have been a nuclear holocaust."

 5.2.3.3  Lewis's Response to the Crucial Objection

         Lewis's response to this objection was set out in his article, "Counterfactual Dependence and Time's Arrow."  The thrust of it is that the theory of counterfactuals in question need not be formulated in terms of our ordinary standards of overall similarity.  Nor should it be, for then the central objection advanced by Bennett and Fine would be correct.  The crucial question, accordingly, is simply whether one can specify weightings for factors that are relevant to the similarity of one world to another that will combine to produce a measure of overall similarity that will generate the right truth-values, and Lewis contends that this is possible.

         In particular, Lewis proposes a weighting of factors according to which, while a perfect match of particular facts for an extended stretch of time counts for more with respect to overall similarity than the absence of a single, small, localized, miracle, or violation of a law of nature, it is less important than the absence of large miracles.  Given this weighting, any world in which Nixon presses the button, if it contains only a small miracle that stops the transmission of the signal, and thus the nuclear holocaust, will not involve a perfect match with respect to future facts­since, for example, the button will be warmer because of contact with Nixon's finger, rays of light in the vicinity of Nixon's hand will be affected differently, Nixon's memories will be different, and so on, all of which will lead to significant divergences with regard to the future state of the world at all later times.  One could, of course, consider a world where all the those differences were eliminated by miracles, but then one would be achieving a perfect match at the cost of a large miracle­a cost that, according to Lewis's proposed criteria for overall similarity, is too high.

 5.2.3.4   The "Nixon and the Button" Objection Revisited

         Lewis's response enabled him to escape the specific formulations of the objection in question that were advanced by Bennett and Fine.  But, as I shall now show, it cannot handle the more general, underlying objection.

         The reason it cannot is that Lewis's proposed solution depends upon the fact that Nixon's pressing the button is an event which has multiple effects, and multiple effects of such a sort that it would take a very big miracle to remove all traces of the event, in order to make it the case that there was a perfect match with the future of the actual world.  But this means that if it is logically possible to construct a parallel case where the crucial event does not have multiple effects of such a sort that it would take anything beyond a small miracle to remove all traces, Lewis's response will not work, and the fundamental objection will stand.

         Can that be done?  The answer is that it can be, and in at least two different ways.  The first involves considering a world that contains at least one type of causal process that has the following two properties:

 (1)  The causal process is non-branching, at least over some temporal interval;

 (2)  The causal process brings about events that are causally necessary conditions for possible subsequent, branching, causal processes.

        Schematically, then, the idea is that one considers a world where there are deterministic causal laws that entail that an event of type C will give rise to a causal process that leads, after a temporal interval t1, to an event of type E, and where, moreover, at every instant during the relevant temporal interval, there is at most one event that is causally related to the event of type C.  If an event of type E occurs, however, in the presence of an event of type D, the result will be multiply-branching causal processes, leading, after a further temporal interval t2, to the occurrence of events of types F1, F2, F3, . . . Fn.  Now let W be a world where such causal laws obtain, but where no event of type C occurs at time t, and consider the following counterfactual:

 (*)  If an event of type C had occurred at time t, then events of types F1, F2, F3, . . . Fn would have occurred at time (t + t1 + t2).

         This counterfactual is clearly true in the world we are considering, but it comes out false on Lewis's approach.  For consider two worlds, W1 and W2, that are otherwise as similar to the original world, W, as possible, but where an event of type C does occur at time t, and where the following propositions are true in the respective worlds:

 In W1:  An event of type E occurs at time (t + t1), and events of types F1, F2, F3, . . . Fn  occur at time (t + t1 + t2);

 In W2:  No event of type E occurs at time (t + t1), and no events of types F1, F2, F3, . . . Fn  occur at time (t + t1 + t2).

         W2 differs from W1 in two respects.  First, it involves a single, small violation of a law of the original world, W, since one has an occurrence of an event of type C at time t, but no event of type E at time (t + t1).  In this respect, W2  is less like the original world than W1  is.  But, secondly, W2  is a perfect match with W from time (t + t1) onward, whereas W1  diverges from W with the occurrence at time (t + t1 + t2) of events of types F1, F2, F3, . . . Fn , and this divergence then becomes ever greater as the resulting causal processes continue to branch.  The upshot is that, given the measure of similarity proposed by Lewis, W2  is closer to the original world, W, than W1  is, and so it follows, on his account of the truth conditions of counterfactuals, that if an event of type C had occurred at time t, neither an event of type E at time (t + t1) nor events of types F1, F2, F3, . . . Fn  at time (t + t1 + t2) would have occurred.  So counterfactual (*) gets wrongly classified as false.

         Formulated in terms of Nixon and the bomb, the example could be as follows. Imagine a world that is different from ours in certain respects.  First, it is a world where it is possible to bring about physical events psychokinetically.  Secondly, it is a world where an act of willing that something be brought about psychokinetically involves no physical change: it consists, instead, only of an appropriate mental state involving emergent qualia.  Finally, such a qualia-state is almost causally impotent: its only effect is the psychokinetically caused occurrence of the event that was willed; there is not even any memory trace of the relevant act of willing in the person who performed the act.

         Here I have formulated things in terms of a direct causal connection between the act of willing and the occurrence of the event willed.  If such a direct connection is thought to be somehow unacceptable, one can easily arrange a mechanism:  there can be a non-branching causal chain that proceeds along a straight line to the location where the event occurs, and where no part of the intervening causal chain has any other effects.

         A strange world, no doubt!  Yet surely one that is logically possible.  But, then, imagine Nixon­or a Nixon counterpart­in a world of this type where he does not will that the button be pressed psychokinetically.  What would be the case if Nixon, in such a world, had willed that the button be pressed psychokinetically?  The correct answer, surely, is given by the following counterfactual:

 "If Nixon had willed that the button be pressed psychokinetically, then that would have happened, and there would have been a nuclear holocaust."

         But on Lewis's approach, this counterfactual will be false.  For an act of willing that something be brought about psychokinetically, in the world that we are considering, will have only one effect:  the occurrence of the event that was willed to happen.  There is no causal branching, and so only a single, small, localized miracle is required to bring it about that although Nixon has willed that the button be pressed psychokinetically, the button is not pressed, and thus, rather than there being a nuclear holocaust, there is, instead, a perfect match with the future of the original world in which Nixon does not will that the button be pressed psychokinetically.  So if a single, small, localized, miracle contributes less to dissimilarity than a perfect match with respect to all future states of affairs contributes to similarity, then it follows that the above counterfactual is false, rather than true.  So Lewis's response fails:  it cannot handle a variation on the Nixon and the button example.

         There is a second way of showing that Lewis's response does not work, since rather than appealing to the possibility of non-branching causal processes, one can appeal instead to the possibility of causal processes that come to an end.

         Schematically, the idea is that one considers a world where the occurrence of an event of type C gives rise to causal processes that spread out in every direction, each of which gives rise, after a distance d, and a temporal interval t1, to an event of type E.  On their own, events of type E have no effects whatsoever.  However an event of type E does have effects when, and only when, it occurs together with an event of type D, and then the result is multiply-branching causal processes, leading, after a further temporal interval t2, to the occurrence of events of types F1, F2, F3, . . . Fn.  Now let W be a world where these causal laws obtain, but where no event of type C occurs at location s at time t.  Assume, moreover, that there is only a single spatial location, at time (t + t1), where an event of type D occurs at distance d from location s.  Consider, then, the following counterfactual:

 (**)  If an event of type C had occurred at time t at location s, events of types F1, F2, F3, . . . Fn would have occurred at time (t + t1 + t2).

         This counterfactual is clearly true in the world we are considering, for the occurrence of an event of type C would initiate causal processes that spread out in every direction, and, as a consequence, an event of type E would occur at time (t + t1) at the one spatial location at that time where an event of type D occurs at distance d from location s.  This, in turn, would result in the occurrence of events of types F1, F2, F3, . . . Fn at time (t + t1 + t2).

         Counterfactual (**) comes out false, however, on Lewis's approach.  For consider two worlds, W1 and W2, that are otherwise as similar to the original world, W, as possible, but where an event of type C does occur at time t at location s, and where the following propositions are true in the respective worlds:

         In W1:  Events of type E occur at time (t + t1) at every location at distance d from location s, including the one such location where an event of type D occurs, and events of types F1, F2, F3, . . . Fn occur at time (t + t1 + t2);

         In W2:  Events of type E occur at time (t + t1) at every location at distance d from location s, except for the one such location where an event of type D occurs, and no events of types F1, F2, F3, . . . Fn occur at time (t + t1 + t2).

         W2 differs from W1 in two respects.  First, it involves a single, small violation of a law of the original world, W, since one has an occurrence of an event of type C at time t, but no event of type E at time (t + t1) at the one location at distance d from location s where an event of type D occurs.  In this respect, W2 is less like the original world, W, than W1 is.  But, secondly, W2 is a perfect match with W at every moment after time (t + t1) onward, whereas W1 diverges from W at every moment from time (t + t1 + t2) onward, in view of the occurrence of events of types F1, F2, F3, . . . Fn at time (t + t1 + t2), and this divergence then becomes ever greater as the resulting causal processes continue to branch.  The upshot is that, given the measure of similarity proposed by Lewis, W2 is closer to the original world, W, than W1 is, and so it follows, on Lewis's account of the truth conditions of counterfactuals, that if an event of type C had occurred at time t, at location s, then there would have been no occurrence either of an event of type E at time (t + t1) at the one location where an event of type D occurs at distance d from location s, or of any subsequent events of types F1, F2, F3, . . . Fn at time (t + t1 + t2).  So counterfactual (**) gets classified, incorrectly, as false.

         The overall conclusion, accordingly, is that Lewis's attempt to answer the most crucial objection to the whole similarity-across-possible-worlds approach to counterfactuals is unsuccessful, since the possibility of worlds that contain either a single non-branching type of causal process, or else branching causal processes that terminate after a finite time, shows that some counterfactuals get assigned the wrong truth-values by a Stalnaker-Lewis account.

 5.2.3.5  Other Objections to a Stalnaker-Lewis Approach

         I now want to turn to some other objections to the attempt to analyze counterfactuals along Stalnaker-Lewis lines.  There are, however, a number of important objections that I shall not discuss, including the following:

 (1)  The Relation Between ‘P & Q’ and ‘P ÅÆ Q’

         On the Stalnaker-Lewis approach to counterfactuals it appears to be true - unless the standards for similarity turn out to be such that there can be cases where world W2 is as similar to W1 as W1 is to itself - that ‘P & Q ‘logically entails ‘P ÅÆ Q’.  Jonathan Bennett argues, however, that this is clearly unsatisfactory, as is shown by cases where ‘P’'s being true makes it very unlikely that ‘Q’ is true, since in such cases ‘P ÅÆ Q’ appears to be false even though ‘P & Q’ is true.

 (2)  The Relation Between  ‘Would' Counterfactuals and Probabilistic Counterfactuals

         Bennett’s argument suggests another objection, which is as follows.  Consider an indeterministic world, and one where (a) 'P' is made true by some state of affairs, S, at time t1, (b) 'Q' is made true by some state of affairs, T, at time t2, (c) there is no state of affairs prior to time t2 that is a causally sufficient condition for the existence of state of affairs T, and, finally, (d) the total state of affairs that existed at the time of state of affairs S made the probability that Q would be true equal to 0.01.  Then, on the one hand, given that ‘P’ and ‘Q’ are both true, it seems that the following counterfactual is true, on a Stalnaker-Lewis approach:

 (i)  "If P were the case, then Q would have been the case"

 On the other hand, in view of (d), it would seem that the following counterfactual must also be true:

 (ii)  "If P were the case, then the probability that Q would be the case would be equal to 0.01."

 But aren’t these two counterfactuals logically incompatible?

 (3)  The "Less Complex Consequent" Objection

 Consider a world where the laws are such that the following counterfactual is true:

 (i)  A ÅÆ (B or C)

 Suppose further, first, that the world is indeterministic, and that, in particular, while the laws entail that if A is the case, then either B or C will be the case, but do not entail that, if A is the case, then B will be the case, or that, if A is the case, then C will be the case, and secondly, that B and C are logically incompatible.  Then it would seem that neither of the following two counterfactuals would be true:

 (ii)  A ÅÆ B

 (iii)  A ÅÆ C

         On Stalnaker’s approach, however, the truth of (i) entails that either (ii) is true or (iii) is true, since Stalnaker’s approach involves the idea that there is always a closest A-world.  This is not so on Lewis’s approach.  Nevertheless, the case still poses a problem for Lewis.  For suppose that the case is one where, first of all, the only difference between the closest A-worlds in which B is true and the closest A-worlds in which C is true is that B is true in the former worlds, and C in the latter, and, secondly, that the truth-maker for 'B' is a much more complex state of affairs, or a temporally more extended state of affairs, than the truth-maker for C.  Then the closest A-worlds in which B is true will be less similar to the original world than are the closest A-worlds in which C is true.  And so on Lewis's approach it will turn out that the counterfactual

 A ÅÆ C

 is true in the world in question.  Then, since C is, by hypothesis, logically incompatible with B, the following counterfactual must also be true:

 A ÅÆ ~B

 But surely this is wrong.

         This consideration can be reinforced, moreover, if one supposes that there is a law of nature that entails not only that, if A is the case, then either B or C is the case, but that the law is a probabilistic one according to which the likelihood that B is the case is very high, and the likelihood that C is the case is very low.

 (4)  The "Agreement with the Actual World" Objection

         This objection was set out by Pavel Tichy in his paper "A Counterexample to the Stalnaker-Lewis Analysis of Counterfactuals” (1976).  Tichy formulates it in terms of a person, Jones, who always wears a hat if it is raining, whereas, if it is sunny, Jones decides, in some random fashion, whether to wear a hat or not.  Suppose, then, that the world we are considering is one where it is raining on a given day, and thus one where Jones wears a hat.  What is the truth value of the following counterfactual:

 "If it had been sunny on the day in question, Jones would have worn a hat."

         Since a sunny-day world where Jones wears a hat will be more similar to the rainy-day world where Jones wears a hat than will a sunny-day world where Jones does not wear a hat - other things being equal - it would seem that the above counterfactual will turn out to be true on a Stalnaker-Lewis approach.  But surely this consequence is unacceptable.  For given that Jones decides via a random process whether to wear a hat if it is sunny, the following 'might' conditional is surely true:

 "If it had been sunny on the day in question, Jones might not have worn a hat."

 And similarly, if Jones decides by some process that has a 50% chance of turning out in either of two ways, then the following probabilistic counterfactual will be true:

 "If it had been sunny on the day in question, the probability that Jones would not have worn a hat would have been 0.5."

         So again, the idea is that one can appeal to one's intuitions about the 'might' counterfactual and about probabilistic counterfactuals to support the idea that the following counterfactual is not true:

 "If it had been sunny on the day in question, Jones would have worn a hat."

         Alternatively, one may simply appeal directly to the intuition that this latter counterfactual is not true.

         The above objections seem to me plausible, and where Lewis has responded to an objection, I believe that one can show either that Lewis’s response is implausible, or else that the objection can be reformulated so that Lewis’s response no longer works.  In the present context, however, the above four objections turn out not really to be relevant, since, as we shall see later, not just any sound objection to a Stalnaker-Lewis-style analysis of counterfactuals will do:  it is crucial that the objections bear are connected with causation in a certain way.  So let us turn to objections that do have such a connection.

 5.2.3.6  The "Simple Worlds" Objection

 5.2.3.6.1  The Case of the Single Particle World

         The objection to a Stalnaker-Lewis approach to counterfactuals is based upon a type of objection which, I have argued elsewhere, applies to any reductionist account of causation, and it involves considering a world that involves only a single particle ­ call it 'M' ­ with no associated fields, gravitational or otherwise .  The question then is what one is to say about the truth values of the following two counterfactuals:

 (a) If solitary particle M had not existed at time t, then it would not have existed at any later time;

 (b)  If solitary particle M had not existed at time t, then it would not have existed at any earlier time.

 The answer is that there are two conclusions that one can draw.  The first is this:

Conclusion 1:  A Stalnaker-Lewis approach entails that the preceding counterfactuals must have the same truth-value.

         Why so?  Simply because, regardless of what factors one takes as relevant to the type of similarity that is crucial for counterfactuals, one will not be able to assign different truth values to (a) and (b) unless one assigns different weight to the temporal location of one of those factors.

         But what prevents one from doing that?  Mightn't one, for example, assign more weight to perfect matches in the past than to perfect matches in the future?

         My answer is, first, that if one did this, then it could, I believe, be shown that the reference to overall simplicity would no longer be doing any real work, and that what one would have is what Lewis refers to in his article "Counterfactual Dependence and Time's Arrow" as 'Analysis 1' - an analysis that is framed in temporal terms.  Secondly, an analysis that assigned a different weight to a factor when it was earlier than the relevant time than when it was later than the relevant time would also, I think, beg the question against time travel and backward causation.  For even if one holds ­ as I do ­ that backward causation and time travel are logically impossible, this is surely not a conclusion that should be built into one's analysis of counterfactuals.

 The other conclusion that one can draw is this:

Conclusion 2:  Lewis's approach entails not only that the preceding counterfactuals must have the same truth-value; it also entails that they are both false, and that the true counterfactual in this situation is instead:

 (c)  If solitary particle M had not existed at time t, then it (or a particle indistinguishable from it) would still have existed at all later times, as well as at all earlier times.

         This is so because Lewis holds that a complete match between worlds with respect to all future events counts more for similarity than a single miracle counts against similarity, and in the single-particle world that we are considering here, it takes only a single, localized, simple miracle to bring it about that a particle just like the one that dropped out of existence at time t exists at all later times.

 5.2.3.6.2  Lewis's Response to the "Simple Worlds" Objection

 Lewis explicitly considers this sort of objection in his article "Counterfactual Dependence and Time's Arrow".  Here is what he says:

 "It might be otherwise if w0  were a different sort of world.  I do not mean to suggest that the asymmetry of divergence and convergence miracles holds necessarily or universally.  For instance, consider a simple world inhabited by just one atom.  Consider the worlds that differ from it in a certain way at a certain time.  You will doubtless conclude that convergence to this world takes no more of a varied and widespread miracle than divergence from it.  This means, if I am right, that no asymmetry of counterfactual dependence prevails at this world."

         A bold response.  One is reminded once again of the saying that one person's modus ponens is another person's modus tollens.  But is this 'outsmarting' maneuver at all plausible?  In the first place, if one holds that there is no asymmetry of counterfactual dependence in the single particle world, then neither will there be any causation, since even if one rejects Lewis's idea that causation can be analyzed counterfactually, it is surely true that the presence of causation entails the presence of an asymmetry of counterfactual dependence.  But is there any causation in the single particle universe?  If one focuses upon causal interaction, it will be tempting to conclude that there isn't.  But here one needs to ask, first, what account one gives of conservation laws:  Are they causal laws or not?  If so, then by assuming that Conservation of Mass is a law in the world we are considering, it will follow that later temporal slices of that universe are causally dependent upon earlier ones.  Secondly, one also needs to ask what account should be given of identity over time.  Isn't a causal analysis of identity over time very plausible?  If so, how can one jettison it in the present case?

         Secondly, Lewis accepts a causal analysis of temporal priority, and of the direction of time.  So if the single particle world has no asymmetry of counterfactual dependence, and thus no causation, then neither can there be any states of affairs that stand in the earlier than relation.  But can the single particle then be something that exists at different times?  Or can it be something that has, on a persistence view, temporal parts?  What can it mean to say that such a world is a temporal world, rather than a world all of whose dimensions are purely spatial ?

         Thirdly, a single particle world can be viewed as having been arrived at by, so to speak, gradually removing things from a complex world.  Thus, consider our own world, and consider a specific electron.  Let t1 and t2 be any two times, where t2 is later than t1.  Then the t2-stage of the electron in question is caused by the t1-stage, and, similarly, the t2-stage of that electron is counterfactually dependent upon the t1-stage.  Now imagine how the world would be if some particle, other than the electron in question, had not existed.  It would still be the case that t2-stage of the electron in question was caused by the t1-stage, and, similarly, that the t2-stage of that electron was counterfactually dependent upon the t1-stage.  So consider the continuation of this process, imagining possible worlds that contain fewer and fewer particles, but still contain the electron in question.  Throughout this enormously long process, it remains true that t2-stage of the electron in question was caused by the t1-stage, and, similarly, that the t2-stage of that electron is counterfactually dependent upon the t1-stage.  But on Lewis's view there is a point - when the next to last particle is eliminated, and one is left only with the solitary electron - when things are suddenly different: no longer is the t2-stage of the electron in question caused by the t1-stage, and, similarly, no longer is the t2-stage of the electron counterfactually dependent upon the t1-stage.  Indeed, according to Lewis's view, time itself disappears, as it is no longer true that the t2-stage of the electron is later than the t1-stage, since the disappearance of causation entails, on a causal theory of time - which Lewis accepts -the disappearance of the earlier than relation.

 5.2.3.7  A Causally Isolated Simple Part in a Very Complex World

         Lewis's very quick dismissal of the single particle world as one that simply does not involve any asymmetry of counterfactual dependence seems, accordingly, very implausible.  But what I now want to argue is that one can modify the case of the single particle world to get another case that constitutes a strong objection to the Stalnaker-Lewis approach to counterfactuals.

         The basic idea is to embed the single particle scenario into a very complex world, as follows.  Consider possible worlds that are rather like ours except for the fact that photons are not affected by gravitational fields.  In some of these worlds there could be a single photon either that was very, very far from everything else, and that never causally interacted with anything else.  Now consider the following two counterfactuals:

 (a) If causally isolated photon M had not existed at time t, then it would not have existed at any later time;

 (b)  If causally isolated photon M had not existed at time t, then it would not have existed at any earlier time.

         We can now draw conclusions precisely parallel to those that were drawn in the case of the very simple universe that contained a single particle.  Thus, in the first place, we have the following conclusion:

Conclusion 1:  The Stalnaker-Lewis approach entails that the preceding counterfactuals must have the same truth-value.

 This obtains because, once again, the Stalnaker-Lewis approach cannot plausibly assign different weights to perfect matches which are the same, but which occur at different times.

         Secondly, we have the following conclusion concerning Lewis's approach:

Conclusion 2:  Lewis's approach entails not only that the preceding counterfactuals must have the same truth-value; it also entails that they are both false, and that the true counterfactual in this situation is instead:

 (c)  If causally isolated photon M had not existed at time t, then it (or a particle indistinguishable from it) would still have existed at all later times, as well as at all earlier times.

And, once again, the reason is that according to Lewis's approach, a complete match between worlds with respect to all future events counts more for similarity than a single miracle counts against similarity, and in the complex world containing a causally isolated particle that we are considering here, it takes only a single, localized, simple miracle to bring it about that a particle just like the isolated one that dropped out of existence at time t exists at all later times.

         A variation on this counterexample in which the particle in question is not always isolated is also possible.  Thus, consider a world where photons are unaffected by gravitational fields, and where there is a photon that interacted with other particles before time t, but that did not do so at time t or afterwards, perhaps because of an unending expansion of the universe.  Then, once again, the following counterfactual -

 If causally isolated photon M had not existed at time t, then it would not have existed at any later time

 - will turn out to be false if one accepts Lewis's measure of similarity, since a single, small miracle, involving the existence of photon qualitatively indistinguishable from photon M at appropriate locations after time will contribute less to dissimilarity of worlds than the perfect match over all future times that is thereby achieved will contribute to similarity.

 5.2.3.8  The Inverted Worlds Objection

         This objection is based upon another objection that I have directed against reductionist approaches to causation.  There the thrust of the objection is that reductionist approaches generate the wrong direction for causal processes in certain rather unusual, but logically possible universes.  Similarly, the thrust here is that the Stalnaker-Lewis approach to counterfactuals generates the wrong direction for counterfactual dependence in the universes in question.

         The basic idea is that there could be worlds that were 'temporally inverted twins'.  To put things concretely, suppose that it is the year 4004 B.C.  A Laplacean-style deity ­ albeit one with more of a sense of humor than many deities ­ is about to create a world rather similar to ours, but one where Newtonian physics is true.  Having created a 4004 B.C. world, and having selected the year 2000 A.D. as a good time for Armageddon, the deity works out what the 4004 B.C. world will be like at that point, down to the last detail.  He then decides to create two spatially unrelated worlds:  the one just mentioned, together with another whose initial state is a 'temporally flipped over' version of the final state of the first world at the time of Armageddon.  That is to say, the final state of the first world agrees exactly with the initial state of the second world, except that the velocities of the particles in the one state are exactly the opposite of the corresponding ones in the other.

 Consider, now, any two complete temporal slices of the first world, A and B, where A is earlier than B.  Since the worlds are Newtonian ones, and since the laws of Newtonian physics are invariant with respect to time reversal, the world that starts off from the reversed, 2000 A.D. type state will go through corresponding states, B* and A*, where these are flipped over versions of B and A respectively, and where B* is earlier than A*.  So while the one world goes from a 4004 B.C., Garden of Eden state to a 2000 A.D., Armageddon state, the other world will move from a reversed, Armageddon type state to a reversed, Garden of Eden type state.

         Let us refer to the two worlds as, respectively, W and W*.  In W, the following counterfactual is true:

 "If state of affairs A had not obtained, then state of affairs B would not have obtained"

         So if, for example, one considers Stalnaker's account, then the closest non-A-world to W - call it V - will be a non-B-world.  But then, in view of the fact that W* is a temporally inverted twin of world W, the temporally inverted twin of world V - call it V* - must have the following properties:

 (a)  V* is a world where state of affairs A* does not obtain;

 (b)  V* is a world where state of affairs B* does not obtain

 (c)  V* is the closest non-A*-world to world W*.

 So it must be case, given a Stalnaker account, that the following counterfactual is true in world W*:

 "If state of affairs A* had not obtained, then state of affairs B* would not have obtained"

         And precisely the same is true if one adopts, instead, Lewis's slightly more complicated account.  But the counterfactual in question is false:  B* is not counterfactually dependent on A*.  Rather, A* is caused by, and so is counterfactually dependent upon, B*.

         In short, the Stalnaker-Lewis approach entails a consequence that is unacceptable, namely, that the following counterfactuals ­ the one in world W, and the other in world W* ­ must have the same truth-value:

 (a) If A had not existed at time t, then B would not have existed at (t + Dt);

 (b)  If A* had not existed at time t, then B* would not have existed at (t - Dt).

 5.2.4  A Possible Response:  Causation and Stalnaker-Lewis Conditionals

         The above objections provide, I suggest, decisive reasons for concluding that a Stalnaker-Lewis-style approach to counterfactuals must be abandoned.  It might then seem to follow that, unless some other approach to counterfactuals that does not involve any causal concepts can be found, approaches to causation of the sort advanced by David Lewis are also doomed.  But, in fact, this is not quite right.

         The reason is this.  Suppose that a Stalnaker-Lewis type of account of the truth conditions of counterfactuals is unsound.  It is still the case that it serves to define a certain conditional - though perhaps one that does not correspond to any conditional found in any natural language.  But there is surely nothing wrong with that, and so it might be suggested that causation can be analyzed, along something like the lines proposed by Lewis, in terms of what can now be viewed as a new conditional ­ the Stalnaker-Lewis conditional.  The resulting analysis of causation will no longer be a counterfactual analysis, of course, but it need not be any the worse for that.

         This is an important objection.  But notice, in the first place, that it is not quite true that nothing is lost if one shifts to the view that the conditionals used in the analysis, rather than being counterfactuals, are some new type of conditional.  For, after all, there do appear to be some conceptual connections between causation and counterfactuals, and even if many philosophers are inclined to think that the source of these connections lies in the fact that causation enters into the analysis of counterfactuals, rather than vice versa, the existence of such connections lends at least some intuitive attraction to the idea of a counterfactual analysis of causation.  By contrast, if Stalnaker-Lewis conditionals, rather than being counterfactuals, are simply a novel type of conditional, then any intuitive basis for a Lewis-style analysis of causation would vanish, and it would then be a remarkable accident indeed if the resulting analysis turned out to be sound.

         In the second place, however ­ and this is the crucial point ­ the objections on which I have focused above also serve to show that causation cannot be analyzed in a Lewis-style way.  For consider, first of all, the case of Nixon and the button, and rather than a world where Nixon does not push the button, consider a world ­ call it W0 ­ where Nixon does psychokinetically push the button, thereby producing an exciting nuclear war.  In that world, what would have happened if Nixon had not willed that the button move?  One possibility is this:

 W1:  The button does not move, and there is no nuclear war, but the world is otherwise as close to W0 as possible.

 Another possibility is this:

 W2:  The button does move, there is a nuclear war, and all events from the time at which the button moves, on into the future, agree completely with those in W0.

         Which of these worlds is closest to W0?  W2 matches W0 perfectly with regard to all future events after a certain time, whereas W1 does not.  On the contrary, the future of W1 is radically different from that of W0. On the other hand, W1 does not involve any miracles at any time after the moment at which Nixon decides not to push the button psychokinetically, whereas W2 does, since the button moves, even though it was not psychokinetically pushed.  But the miracle in question is only a single, small miracle, and, according to Lewis’s criteria for similarity of the relevant sort, the presence of a complete match of W2’s future with that of W0 is a more significant factor than the fact that W2 involves a single, small miracle, while W1 does not.  Therefore, world W2 is closer to world W0 than world W1 is.  Consequently, the following Stalnaker-Lewis conditional will be true in world W0:

 If Nixon had not willed that the button be pressed psychokinetically, the button would have moved.

         Thus the movement of the button is not Stalnaker-Lewis, conditionally dependent upon Nixon’s willing that the button be pressed psychokinetically, and since such dependence lies at the very heart of the approach to causation that we are considering, it follows from the truth of the above conditional that Nixon’s willing that the button be pressed psychokinetically does not, in world W0, cause the button to move.  But, by hypothesis, it does.  Therefore causation cannot be analyzed in terms of the novel, Stalnaker-Lewis conditionals.

         The other objections on which I focused lead to the same conclusion.  Consider, for example, the case of the isolated particle in the very complex world.  Whatever measure of similarity one chooses, the two conditionals we considered earlier will have the same truth-values, and so it will be the case either that later temporal stages of the particle are not causally dependent upon earlier ones, or that they are, but that it is also the case that earlier temporal stages are also causally dependent upon later ones, and neither consequence seems acceptable.  In addition, if one adopts the measure of similarity that Lewis advances, it will turn out that if the particle had not existed at any moment, it would still have existed at later moments, and so later stages of the particle turn out not to be causally dependent upon earlier stages.

         Finally, consider the inverted world case.  Whatever measure of similarity between worlds one chooses, if that measure generates Stalnaker-Lewis, conditional dependence that runs in the right direction in the non-inverted world, it will generate conditional dependence that runs in the opposite direction to that of causation in the inverted world.  So once again, any analysis of causation in terms of Stalnaker-Lewis conditionals cannot be sound.

 5.2.5  Summing Up:  Counterfactual Analyses of Causation

         A number of philosophers have advanced some very important objections to counterfactual analyses of causation ­ objections that I surveyed very briefly at the beginning of this paper.  These include:  (1) the dependence of Lewis’s analysis upon an account of the individuation of events;  (2) the existence of counterfactuals that have nothing to do with causation, or even with laws of nature;  (3) counterfactuals that are based upon non-causal laws;  (4) situations involving causal overdetermination;  and (5) cases of preemption, including ‘trumping’ preemption.

         All of these objections pose serious, prima-facie obstacles for any counterfactual analysis of causation.  It may well be, however, that some of them can be surmounted by an appropriately formulated account.  Thus I am inclined to think, for example, that difficulties concerning the right account of the individuation of events can be eliminated simply by adopting the view that causal relata are states of affairs, rather than events.  On the other hand, some of the other objections seem much more problematic, and, in particular, it seems to me very doubtful that there is any satisfactory way of handling the causal overdetermination objection.

         The basic thrust of this discussion, however, has been that there is a more fundamental flaw in the whole idea of a counterfactual analysis of causation, and one which cannot be avoid by any tinkering with the details of the analysis.  This is, first, that a counterfactual analysis requires an account of the truth conditions of counterfactuals that does not itself involve any causal concepts, and secondly, that there are excellent reasons for thinking that no adequate, non-causal account of counterfactuals is at all likely to be forthcoming, since the only serious candidate for such an account ­ namely, a Stalnaker-Lewis-style analysis of counterfactuals ­ appears to be open to a number of decisive objections.

         Finally, I considered the idea that one might abandon the project of a counterfactual analysis of causation, but hold that causation can still be analyzed in terms of Stalnaker-Lewis-style conditionals, where the latter are no longer identified with counterfactual conditionals.  Such a move would, I noted, deprive the proposed analysis of any intuitive basis.  But, more importantly, one can show that a number of central objections to a Stalnaker-Lewis approach to counterfactuals also tell against any attempt to analyze causation in terms of new, non-counterfactual conditionals defined along Stalnaker-Lewis lines.

 5.3  Probabilistic Approaches:  Causation and Relative Frequencies

         Through the end of the 19th Century, almost all philosophers thought of causation as connected with conditions that were totally sufficient to ensure the occurrence of an event.  But that changed in the 20th Century, with the emergence of quantum physics, and the development of the social sciences, and many philosophers gradually came to feel that causation is not restricted to cases where there are causally sufficient conditions for the occurrences of events.

         What implications does this have for the philosophy of causation?  One possible view is that it has very little relevance.  For while quantum physics certainly appears to provide excellent reason for holding that causation can be present in situations that do not fall under deterministic laws, this need not imply that one's concept of causation has to be revised.  Perhaps all that is needed is a concept of probabilistic laws, a concept which can then be combined with one's prior concept of causation to generate a satisfactory account of causation in probabilistic settings.

         But there is also a very different possibility that needs to be explored:  perhaps the right route involves an account of causation that is itself genuinely probabilistic, so that the concept of probability, rather than merely entering via probabilistic laws, is part of the very analysis of the relation of causation itself.

 5.3.1  The Basic Approach

         The earliest attempts to formulate a probabilistic analysis of causation were advanced by Hans Reichenbach (1956), I. J. Good (1961 and 1962), and Patrick Suppes (1970), and they all were based upon the idea of probability understood in terms of relative frequency.  Moreover, no use was made of the idea of laws of nature, let alone of a realist conception of laws, so that what Reichenbach, Good, and Suppes offered were strong reductionist accounts of causation, and ones that involved only Humean states of affairs.

         At the heart of any probabilistic analysis of causation is the idea that causes must, in some way, make their effects more likely, and within these initial probabilistic accounts of causation the basic idea was to analyze what it is for a cause to make its effect more likely in terms of the notion of positive statistical relevance, where an event of type B is positively relevant to an event of type A if and only if the conditional probability of an event of type A relative to an event of type B is greater than the unconditional probability of an event of type A.   Thus Suppes (1984, p. 151), for example, introduces the notion of a prima facie cause, defined as follows: "An event B is a prima facie cause of an event A if and only if (i) B occurs earlier than A, and (ii) the conditional probability of A occurring when B occurs is greater than the unconditional probability of A occurring."

         Perhaps the most crucial test for any theory of causation is whether it can provide a satisfactory account of the direction of causation.  What account can be offered, given a probabilistic approach?  One possibility, of course, is to incorporate the earlier than relation into one's analysis of causation, and to use that relation to define the direction of causation -- as was done, for example, by Suppes (1970).

         It is widely thought, however, that this is not satisfactory.  One reason is that it then follows immediately both that it is logically impossible for a cause and its effect to be simultaneous, and for a cause to be later than its effect, and while both things may be the case, the fact that many people have thought, for example, that time travel into the past is logically possible surely provides good reason for holding that it cannot be an immediate consequence of the analysis of causation that backward causation is logically impossible.

         Another consideration is that there is a serious problem about what it is that is the basis of the direction of time, and a causal theory of time has been thought by many philosophers to be a possibility worthy of serious consideration.  If so, then the direction of causation cannot be defined in terms of the direction of time.

         Because of considerations such as these, most advocates of a probabilistic approach to causation have wanted to analyze the direction of causation in probabilistic terms.  What are the prospects for doing this?  The first thing to note is that the postulate that a cause raises the probability of its effect does not itself provide any direction for causal processes.  For when the following equation for conditional probabilities

 Prob(E/C) x Prob(C)  =  Prob(E & C)  =  Prob(C/E) x Prob(E)

 is rewritten as

 Prob(E/C)/Prob(E)  =   Prob(C/E)/Prob(C)

 one can see that Prob(E/C) > Prob(E)  if and only if  Prob(C/E) > Prob(C)

 So causes raise the probabilities of their effects only if effects also raise the probabilities of their causes.

         How, then, can the direction of causation be analyzed probabilistically?  The most promising suggestion was set out by Reichenbach in his book The Direction of Time (1956).  Reichenbach's proposal involves the following elements:  first, what he referred to as 'the Principle of the Common Cause'; secondly, a probabilistic characterization of a 'conjunctive fork'; thirdly, a proof that correlations between event-types can be explained via conjunctive forks; and, fourthly, a distinction between open forks and closed forks.

         As regards the first element, Reichenbach's Principle of the Common Cause is as follows:  'If an improbable coincidence has occurred, there must exist a common cause.'  (Reichenbach, 1956, p. 157)  Here the basic claim is that if events of type A, say, are more likely to occur given events of type B, than in the absence of events of type B, and if the explanation of this is not that events of type A are caused by events of type B, or vice versa, then there must some third type of event -- say, C -- such that events of type C cause both events of type A and events of type B.

         Secondly, there is Reichenbach's characterization of the idea of a conjunctive fork, which - using a slightly different notation - can be set out as follows (1956, p. 159):

 Events of types A, B, and C form a conjunctive fork if and only if:

 (1)  Prob(A & B/C) = Prob(A/C) x Prob(B/C)

 (2)  Prob(A & B/not-C) = Prob(A/not-C) x Prob(B/not-C)

 (3)  Prob(A/C) > Prob(A/not-C)

 (4)  Prob(B/C) > Prob(B/not-C)

         Thirdly, Reichenbach then shows that, provided that none of the relevant probabilities is equal to zero, equations (1) through (4) entail:

 (5)  Prob(A & B) > Prob(A) x Prob(B)

 This in turn entails:

 (6)  Prob(A/B) > Prob(A)

 (7)  Prob(B/A) > Prob(B)

     So we see that the existence of a conjunctive fork involving event-types A, B, and C provides an explanation of a statistical correlation between the event-types A and B.

         Finally, Reichenbach then distinguishes between open forks and closed forks.  Suppose that events of types A, B, and C form a conjunctive fork, and that there is no other type of event -- call it E -- such that events of types A, B, and E also form a conjunctive fork.  Then A, B, and C form an open fork.  On the other hand, if there is another type of event, E, such that events of types A, B, and E also form a conjunctive fork, what one has is a closed fork.

         As Reichenbach emphasizes, there can certainly be conjunctive forks that involve common effects, rather than common causes. (1956, pp. 161-2).  But since conjunctive forks can, as we have just seen, explain statistical correlations, if there were an open fork that involved a common effect, then the relevant statistical correlation would be explained, even though there was no common cause, and this would violate the Principle of the Common Cause.  Hence, conjunctive forks involving a common effect must, if Reichenbach is right, always be closed forks.  All open forks, therefore, must involve a common cause, and so the direction of causation is fixed by the direction given by open forks.

 5.3.2  Objections

         This is a subtle and ingenious attempt to offer a probabilistic analysis of the relation of causation, and one that appeals only to Humean states of affairs.  Unfortunately, it appears to be open to a number of decisive objections.

 5.3.2.1  Accidental, Open Forks Involving Common Effects

         The basic idea here is simply this.  Suppose that A and B are types of events that do not cause one another, and for which there is no common cause.  Then it might be the case that the conditional probability of an event of type A given an event of type B was exactly equal to the unconditional probability of an event of type A, but surely this is not necessary.  Indeed, it would be more likely that the two probabilities were at least slightly different, so that the conditional probability of an event of type A given an event of type B was either greater than or less than the unconditional probability of an event of type A.

         Let us suppose, then, that the third of these alternatives is the case.  Suppose, further, that the occurrence of an event of type A is a causally necessary condition for the occurrence of a slightly later event of type E, and, similarly, that the occurrence of an event of type B is a causally necessary condition for the occurrence of a slightly later event of type E.

         Finally, let us suppose -- as is perfectly compatible with the preceding assumptions -- that the relative numbers of all possible combinations of events of types A, B, and E, throughout the whole history of the universe, are given by the following table:

                                                   E                                             Not-E

                                          A          Not-A                       A                      Not-A

 B                                      1             0                           18                          12

 Not-B                              0              0                           42                          28

         From this table, one can see that Prob(A) = 61/101, or about .604, while Prob(A/B) = 19/31, or slightly less than .613, so that, if the absolute numbers are not too large, there will be nothing especially remarkable about the fact that Prob(A/B) > Prob(A)

         Next, examining the numbers that fall under 'E', we can see that we have the following probabilities:

 Prob(A/E) = 1;  Prob(B/E) = 1;  Prob(A & B/E) = 1.

 Hence the following is true:

 (1)  Prob(A & B/E) = Prob(A/E) x Prob(B/E)

         Similarly, examining the numbers that fall under 'Not-E', we can see that we have the following probabilities:

 Prob(A/Not-E) = 60/100 = 0.6;  Prob(B/Not-E) = 30/100 = 0.3;  Prob(A & B/E) = 18/100 = 0.18.

         So the following three equations are also true:

 (2)  Prob(A & B/not-E) = Prob(A/not-E) x Prob(B/not-E)

 (3)  Prob(A/E) > Prob(A/not-E)

 (4)  Prob(B/E) > Prob(B/not-E)

         Hence, in a universe of the sort just described, the three types of events A, B, and E form a conjunctive fork.  Moreover, since there is, by hypothesis, no type of event, C, that is a common cause of events of types A and B, it is therefore the case that A, B, and E constitute an open fork.  This open fork then defines the relevant direction of causation as the direction that runs from events of type E towards events of the two types, A and B, that are causally necessary conditions for the occurrence of an event of type E.

         In short, not only is it logically possible to have an open fork that involves a common effect, rather than a common cause, but there is no significant unlikelihood associated with the occurrence of such an open fork.  The direction of open forks cannot, therefore, serve to define the direction of causation.

 5.3.2.2  Underived Laws of Co-Existence

         John Stuart Mill suggested that, in addition, to causal laws, there could be basic laws of necessary co-existence that related simultaneous states of affairs.  Are such laws possible?  If one considers some candidates that might be proposed, it may be tempting, I think, to be attracted to the idea that although there can be laws of necessary co-existence, all such laws are derived, rather than basic, though this idea is far from unproblematic.  Thus, consider, for example, a Newtonian world, and Newton's Third Law of Motion -- that if one body, X exerts a certain force, F, on another body Y, then Y exerts a force equal in magnitude to F, and opposite in direction, upon X.  This certainly asserts the existence of a necessary connection between simultaneous states of affairs, but is it correctly viewed as a basic law, in a Newtonian universe?  Doubts arise, I think, in view of the fact that the fundamental force laws entail conclusions such as the following:

 (a)  It is a law that for any objects X and Y, and any time t, if X exerts a gravitational force F on Y at time t, then Y exerts a gravitational force -F on X at time t.

 (b)  It is a law that for any objects X and Y, and any time t, if X exerts an electrostatic force F on Y at time t, then Y exerts a gravitational force -F on X at time t.

 (c)  It is a law that for any objects X and Y, and any time t, if X exerts a magnetic force F on Y at time t, then Y exerts a gravitational force -F on X at time t.

         So non-causal laws that are special instances of Newton's Third Law of Motion can be derived from the fundamental force laws, and the latter are, if one treats forces realistically, causal laws.

         But this is not, of course, a derivation of Newton's Third Law of Motion itself.  To have the latter, it would have to be the case that there was a law to the effect that there were only certain types of forces:  gravitational, electrostatic, magnetic, etc.  Moreover, even if the latter were a law in a Newtonian universe, it would not be a causal law, and so one would still not have a derivation of Newton's Third Law of Motion from causal laws alone.

         It is, accordingly, far from clear that Newton's Third Law of Motion can be derived from causal laws.  But a philosopher who wishes to maintain that all basic laws are causal laws has a different response available -- namely that, in a Newtonian universe, Newton's Third Law of Motion would not really be a law in the strict sense: it would be, instead, a generalization based upon the forces and force laws that have been discovered to this point.
 
         Whether this response is ultimately correct, I do think that it shows at least that it is unclear whether Newton's Third Law of Motion would be a case of a basic, non-causal law of co-existence.  But even if this particular example is doubtful, how can one rule out the possibility of there being such laws?  Why could it not be a law, for example, that all particles with mass M have charge C, and vice versa, without that law's being derivable from any other laws whatever?  The claim that this is not possible surely requires an argument.  But what could the argument possibly be?

         In the absence of a proof of the impossibility of basic, non-causal laws of co-existence, it seems to me that one is justified in holding that such laws are logically possible.  But if this is right, then Reichenbach's Principle of the Common Cause is unsound, since the extremely improbable coincidence that all particles with mass M have charge C, and vice versa, rather than being explained causally, might simply obtain in virtue of a basic, non-causal law.

 5.3.2.3  Underived Laws of Co-Existence, and Non-Accidental, Open Forks Involving Common Effects

         If there can be such laws, that also allows one to show that there can be open forks involving common effects that, rather than depending upon accidents of distribution, arise simply in virtue of certain laws.  In particular, consider a world in which the following things are the case:

 (a)  The occurrence of an event of type A is a causally necessary condition for the occurrence of a slightly later event of type E;

 (b)  The occurrence of an event of type B is a causally necessary condition for the occurrence of a slightly later event of type E;

 (3c  The co-occurrence of an event of type A and an event of type B is a causally sufficient condition for the occurrence of a slightly later event of type E;

 (d)  It is a basic, non-causal law that an event of type A is always accompanied by an event of type B, and vice versa.

         Then, provided that there is at least one occurrence of an event of type E, the following probabilities must obtain in virtue of (a) through (d):

 Prob(A/E) = 1;  Prob(B/E) = 1;  Prob(A & B/E) = 1.

 Prob(A/not-E) = 0;  Prob(B/not-E) = 0;  Prob(A & B/not-E) = 0.

         It then follows that the following four equations are all true:

 (1)  Prob(A & B/E) = Prob(A/E) x Prob(B/E)

 (2)  Prob(A & B/not-E) = Prob(A/not-E) x Prob(B/not-E)

 (3)  Prob(A/E) > Prob(A/not-E)

 (4)  Prob(B/E) > Prob(B/not-E)

         So the conclusion, accordingly, is that if there can be basic laws of co-existence, then there can be cases of open forks involving common effects that obtain, not by accident, but in virtue of laws of nature.

 5.3.2.4  Simple, Deterministic, Temporally Symmetric Worlds

         The next objection to the present probabilistic analysis of causation applies to any reductionist account of a Humean sort, and the basic idea is this.  On the one hand, the actual world is a complex one, with a number of features that might be invoked as the basis of a reductionist account of the direction of causation.  For, first of all, the direction of increase in entropy is the same in the vast majority of isolated or quasi-isolated systems (Reichenbach, 1956, pp. 117-43, and Adolf Grünbaum, 1973, pp. 254-64).  Secondly, the temporal direction in which order is propagated -- such as by the circular waves that result when a stone strikes a pond, or by the spherical wave fronts associated with a point source of light -- is invariably the same (Karl Popper 1956, p. 538 ).  Thirdly, it is also a fact that all, or virtually all, open forks are open in the same direction -- namely, towards the future (Reichenbach, 1956, pp. 161-3, and Wesley Salmon, 1978, p. 696).

         On the other hand, causal worlds that are much simpler than our own, and that lack such features, are surely possible.  In particular, consider a world that contains only a single particle, or a world that contains no fields, and nothing material except for two spheres connected by a rod, that rotate endlessly about one another, on circular trajectories, in accordance with the laws of Newtonian physics.  In the first world, there are causal connections between the temporal parts of the single particle.  In the second world, each sphere will undergo acceleration of a constant magnitude, due to the force exerted on it by the connecting rod.  So both worlds certainly contain causal relations.  But both worlds are also utterly devoid of changes of entropy, of propagation of order, and of all causal forks, open or otherwise.  The probabilistic analysis that we are considering, however, defines the direction of causation in terms of open forks.  Simple worlds such as those just mentioned show, therefore, that that probabilistic analysis cannot be sound.

         But what if the advocate of such an analysis responded by challenging the claim that such worlds contain causation?  In the case of the rotating spheres world, this could only be done by holding that it is logically impossible for Newton's Second Law of Motion to be a causal law, while in the case of the single particle world, one would have to hold that identity over time is not logically supervenient upon causal relations between temporal parts.  But both of theses claims, surely, are very implausible.

         In addition, however, such a challenge would also involve a rejection of the following principle:

 The Intrinsicness of Causation in a Deterministic World

If C1 is a process in world W1, and C2 a process in world W2, and if C1 and C2 are qualitatively identical, and if W1 and W2 are deterministic worlds with exactly the same laws of nature, then C1 is a causal process if and only if C2 is a causal process.

         For consider a world that differs from the world with the two rotating spheres by having additional objects that enter into causal interactions, and one of which collides with one of the spheres at some time t.  In that world, the process of the spheres rotating around one another during some interval when no object is colliding with them will be a causal process.  But then, by the above principle, the rotation of the spheres about one another, during an interval of the same length, in the simple universe, must also be a causal process.

         But is the Principle of the Intrinsicness of Causation in a Deterministic World correct?  Some philosophers have claimed that it is not.  In particular, it has been thought that a type of causal situation to which Jonathan Schaffer (2000, pp. 165-81) has drawn attention -- cases of 'trumping preemption' -- show that the above principle must be rejected.

         Here is a slight variant on a case described by Schaffer.  Imagine a magical world where, first of all, spells can bring about their effects via direct action at a temporal distance, and secondly, earlier spells prevail over later ones.  At noon, Merlin casts a spell to turn a certain prince into a frog at midnight -- a spell that is not preceded by any earlier, relevant spells.  A bit later, Morgana also casts a spell to turn the same prince into a frog at midnight.  Schaffer argues, in a detailed and convincing way, that the simplest hypothesis concerning the relevant laws entails that the prince's turning into a frog is not a case of causal overdetermination: it is a case of preemption.

     It differs, however, from more familiar cases of preemption, where one causal process preempts another by preventing the occurrence of some event that is crucial to the other process.  For in this action-at-a-temporal-distance case, both processes are fully present, since they consist simply of the casting of a spell plus the prince's turning into a frog at midnight.

         A number of philosophers, including David Lewis (2000), have thought that the possibility of trumping preemption shows that the Principle of the Intrinsicness of Causation in a Deterministic World is false, the idea being that there could be two qualitatively identical processes, one of which is causal and the other not.  For example, at time t1, Morgana casts a spell that a person turn into a frog in one hour's time at a certain location  That person does turn into a frog, because there was no earlier, relevant spell.  At time t2, Morgana casts precisely the same type of spell.  The person in question does turn into a frog, but the cause of this was not Morgana's spell, but an earlier, preempting spell.

         Is this a counterexample to the Intrinsicness Principle?  The answer is that it is not.  Causes are states of affairs, and the state of affairs that, in the t1 case, causes the person to turn into a frog is not simply Morgana's casting of the spell:  it is that state of affairs together with the absence of earlier, relevant spells.  So when the complete state of affairs that is the cause is focused upon, the two spell-casting cases are not qualitatively identical.  Trumping preemption is therefore not a counterexample to the Principle of the Intrinsicness of Causation in a Deterministic World.

 5.3.2.5  Simple, Probabilistic, Temporally Non-Symmetric Worlds

         The two simple, possible worlds mentioned in the preceding section were deterministic worlds, and they were also worlds that, as regards non-causal states of affairs, were precisely the same in both temporal directions.  Because of the latter property, they are counterexamples to any Humean, reductionist analysis of causation.  For given the complete temporal symmetry, there cannot be any Humean feature that will serve to pick out one of the two temporal directions as the direction of causation.

         That complete temporal symmetry also meant, however, that there is no evidence in such worlds as to what the direction of causation is, and, for those with verificationist tendencies, this will be viewed as a reason for denying that there is any direction to causation in those worlds.  What I now want to do, accordingly, is to show that there are other simple worlds that are equally counterexamples to Humean, reductionist analyses of causation, but that are not temporally symmetric, and that, because of the precise way in which they are asymmetric, are worlds that contain very strong evidence concerning the likely direction of causation.

         Consider a world that contains states of affairs of types S0(x, t), S1(x, t), S2(x, t), S3(x, t), . . . Sn(x, t), which are as follows.  First, S0(x, t) is a states of affair in which absolutely nothing exists at  location x at time t.  Secondly, if i is odd, Si(x, t), consists of 2i atomic elements of type A that are equally spaced on a circle of radius r, while if i is even, Si(x, t), consists of 2i atomic elements of type B that are equally spaced on a circle of radius r, .  So, leaving aside the circular arrangement of elements, the first few states of affairs are as follows:

 S0(x, t):         Nothing at all

 S1(x, t):         A  A

 S2(x, t):         B  B  B  B

 S3(x, t):         A  A  A  A  A  A  A  A

 S4(x, t):         B  B  B  B  B  B  B  B  B  B  B  B  B  B  B  B

 S5(x, t):         A  A  A  A  A  A  A  A  A  A  A  A  A  A  A  A A  A  A  A  A  A  A  A   A  A  A A  A  A  A  A

         Consider, now, the following two possible laws:

 L1:  For every region x, and every time t, if there is a state of affairs of type Si(x, t), where i is greater than 0, and less than n, that state of affairs will continue to exist until it has existed for a temporal interval of length d, at which point it will be replaced by a state of affairs of type Si+1(x, t*), where the spatial orientation of the latter state of affairs with respect to that of the temporally preceding one is completely random, while, if there is a state of affairs of type Sn(x, t), that state of affairs will continue to exist until it has existed for a temporal interval of length d, at which point it will be replaced by a state of affairs of type S0(x, t*).

 L2:  For every region x, and every time t, if there is a state of affairs of type Si(x, t), where i is greater than 0, that state of affairs will continue to exist until it has existed for a temporal interval of length d, at which point it will be replaced by a state of affairs of type Si-1(x, t*), where the spatial orientation of the latter state of affairs with respect to that of the temporally preceding one is completely random.

         Why have I specified that it is a completely random matter how successive states of affairs are spatially oriented relative to one another?  The answer is that this has been done to make it impossible, given the present account of causation, for there to be any causal forks in any world whose only law is either L1 or L2.  For consider the transition from S1 to S2.  If the relative spatial orientation of S1 and S2 is a random matter, then there is nothing that can make it the case, given the present account, that one of the two A elements in the state of type S1 is causally related to two specific B elements in the succeeding S2 state.  All that one will be able to say is that the one total state of affairs causes the other total state of affairs, and because one cannot break this down into relations between parts of one and parts of the other, no causal forks will exist.

         Suppose now that T1 and T2 are two types of worlds, each with the same, very large number of spatial locations.  Suppose, further, that L1 is the only law in worlds of type T1, and that L2 the only law in worlds of type T2, and that in worlds of type T1,  a state of affairs of type S1(x, t) sometimes pops into existence, completely uncaused, in vacant regions of sufficient size, while, in worlds of type T2, a state of affairs of type Sn(x, t) sometimes pops into existence, completely uncaused, in vacant regions of sufficient size.

         Suppose, finally, that W is a world that is either of type T1 or of type T2.  As we have seen, because it is a completely random matter how successive states of affairs are spatially oriented relative to one another, there cannot be, given the probabilistic analysis of causation that we are now considering, any forks in world W -- and, a fortiori, any open forks.  It therefore follows, on this analysis of causation, that there is no direction of causation, and so no causation, in world W.

         But this conclusion is unsound.  The information that one has about the world makes it very likely that there is causation in world W, and that it has a certain direction.  For compare worlds of type T1 with worlds of type T2.  In worlds of the former sort, the only type of state of affairs that comes into existence uncaused is a state of affairs of type S1(x, t), and since this consists of only two atomic elements of type A, it is not especially unlikely that such a state of affairs should come into existence uncaused.  By contrast, in worlds of type T2, the type of state of affairs that comes into existence uncaused is a state of affairs of type Sn(x, t), and since this may very well consist of an enormous number of atomic elements -- since n can be any number one wants, such as 10100 -- all of them of the same type, equally spaced on a circle, it may, by contrast, be extraordinarily unlikely that such a state of affairs should come into existence uncaused.

         The upshot, in short, is that given a world W that is either of type T1 or of type T2, it is much more likely that W is of type T1 than of type T2, and so it is much more likely that the direction of causation runs from states of affairs of type S1(x, t) to type S2(x, t) and on to type Sn(x, t), than that it runs in the opposite direction.

         Finally, though worlds of types T1 and T2 do involve laws that are not completely probabilistic, since the temporal interval at which one state of affairs is replaced by another is fixed, that it not essential, and one could replace laws L1 and L2 by totally probabilistic laws in which each of the relevant states of affairs has a certain half-life, so that there would merely be a certain probability that a given state of affairs would, within a given temporal interval, be replaced by the next state in the relevant order.  The resulting world types - T1* and T2* - would then be completely probabilistic worlds, but that would not alter the fact that it would be much more likely that the direction of causation was from states of affairs of type S1(x, t) to states of affairs of type S2(x, t) and on to states of affairs of type Sn(x, t), rather than in the opposite direction.

         The conclusion, accordingly, is that there are simple, probabilistic worlds in which causation is present, and in which there is good reason for viewing one of the two possible temporal directions as the direction of causation, but where the probabilistic analysis of causation that we are considering mistakenly entails that no causation is present.

 5.3.2.6  Temporally 'Inverted', Twin Universes

         It is the year 4004 B.C.  A Laplacean-style deity is about to create a world rather similar to ours, but one where Newtonian physics is true.  Having selected the year 3000 A.D. as a good time for Armageddon, the deity works out what the world will be like at that point, down to the last detail.  He then creates two spatially unrelated worlds:  the one just mentioned, together with another whose initial state is a flipped-over version of the state of the first world immediately prior to Armageddon - i.e., the two states agree exactly, except that the velocities of the particles in the one state are exactly opposite to those in the other.

         Consider, now, any two complete temporal slices of the first world, A and B, where A is earlier than B.  Since the worlds are Newtonian ones, and since the laws of Newtonian physics are invariant with respect to time reversal, the world that starts off from the reversed, 3000 A.D. type state will go through corresponding states, B* and A*, where these are flipped-over versions of B and A respectively, and where B* is earlier than A*.  So while the one world goes from a 4004 B.C., Garden of Eden state to a 3000 A.D., pre-Armageddon state, the other world will move from a reversed, pre-Armageddon type of state to a reversed, Garden of Eden type of state.

         In the first world, the direction of causation will coincide with such things as the direction of increase in entropy, the direction of the propagation of order in non-entropically irreversible processes, and the direction defined by most open forks.  But in the second world, where the direction of causation runs from the initial state created by the deity -- that is, the flipped-over 3000 A.D. type of state -- through to the flipped-over 4004 B.C. type of state, the direction in which entropy increases, the direction in which order is propagated, and the direction defined by open forks will all be the opposite one.  So if any of the latter is used to define the direction of causation, it will generate the wrong result in the case of the second world.  The probabilistic analysis of causation that we are presently considering assigns, therefore, the wrong direction to causation in the case of the second world.

 5.3.2.7  Causally Ambiguous Situations in Probabilistic Worlds

         A reductionist analysis of causation in terms of relative frequencies is also exposed to a variety of 'underdetermination' objections, the thrust of which is that fixing all of the non-causal properties of, and relations between, events, including all relative frequencies, does not always suffice to fix what causal relations there are between events.  Indeed, the arguments in question support much stronger conclusions -- such as, for example, the conclusion that even if one also fixes what laws there are, both causal and non-causal, along with the direction of causation for all possible causal relations that might obtain, that still does not suffice to settle what causal relations there are between events.

         One such argument can be set out as follows.   First, one needs to ask whether statements of causal laws can involve the concept of causation.  Consider, for example, the following statement:  "It is a law that for any object x, the state of affairs that consists of x's having property F causes a state of affairs that consists of x's having property G."  Is this an acceptable way of formulating a possible causal law?

         Some philosophers contend that it is not, and that the correct formulation is, instead, along the following lines:

 (*)  "It is a causal law that for any object x, if x has property F at time t, then x has property G at (t + *t)."

         But what reason is there for thinking that it is the latter type of formulation that is correct?  Certainly, as regards intuitions, there is no reason why there should not be laws that themselves involve the relation of causation.  But in addition, the above claim is open to the following objection.  First, the following two statements are logically equivalent:

 (1)  For any object x, if x has property at time t, then x has property G at (t + *t);

 (2)  For any object x, if x lacks property G  at time (t + *t), then x lacks property F at t.

         Now replace the occurrence of (1) in (*) by an occurrence of (2), so that one has:

 (**)  "It is a causal law that for any object x, if x lacks property G  at time (t + *t), then x lacks property F at time t."

         The problem now is that it may very well be the case that while (*) is true, (**) is false, since its being a causal law that for any object x, if x has property at time t, then x has property G at (t + *t) certainly does not entail that there is a backward causal law to the effect that for any object x, if x lacks property G  at time (t + *t), then x lacks property F at t.  So anyone who holds that (*) is the correct way to formulate causal laws needs to explain why substitution of logically equivalent statements in the relevant context does not preserve truth.

         By contrast, no such problem arises if one holds that causal laws can instead be formulated as follows:

 It is a law that for any object x, the state of affairs that consists of x's having property F at time t causes a state of affairs that consists of x's having property G at time (t + *t).

         Let us assume, then, that the natural way of formulating causal laws is acceptable.  The next step in the argument involves the assumption that probabilistic laws are logically possible.  Given these two assumptions, the following presumably expresses a possible causal law:

 L1:  It is a law that, for any object x, x's having property P for a time interval *t causally brings it about, with probability 0.75, that x has property Q.

         The final crucial assumption is that it is logically possible for there to be uncaused events.

         Given these assumptions, consider a world, W, where objects that have property P for a time interval *t go on to acquire property Q 76 percent of the time, rather than 75 percent of the time, and that this occurs even over the long term.  Other things being equal, this would be grounds for thinking that the relevant law was not L1, but rather:

 L2:  It is a law that, for any object x, x's having property P for a time interval *t causally brings it about, with probability 0.76, that x has property Q.

         But other things might not be equal.  In the first place, it might be the case that L1 was derivable from a very powerful, simple, and well-confirmed theory, whereas L2 was not.  Secondly, one might have excellent evidence that there were totally uncaused events involving objects' acquiring property Q, and that the frequency with which that happened was precisely such as would lead to the expectation, given law L1, that situations in which an object had property P for a time interval *t would be followed by the object's acquiring property Q 76 percent of the time.

         If that were the case, one would have reason for believing that, on average, over the long term, of the 76 cases out of a 100 where an object that has property P for *t and then acquires property Q, 75 of those cases will be ones where the acquisition of property Q is caused by the possession of property P, while one out of the 76 will be a case where property Q is spontaneously acquired.

         There can, in short, be situations where there would be good reason for believing that not all cases where an object has property P for an interval *t, and then acquires Q, are causally the same.  There is, however, no hope of making sense of this given a reductionist analysis of causation in terms of relative frequencies.  For the cases do not differ with respect to any non-causal properties and relations, including relative frequencies, nor with respect to causal or non-causal laws, nor with respect to the direction of causation in any potential causal relations.   So the present approach is unable to deal with such causally ambiguous, probabilistic situations.

 5.3.2.8  Causation Without Increase in Probability

         We have not yet considered the most fundamental claim involved not only in the attempt to analyze causation in terms of relative frequencies, but, indeed, in all probabilistic analyses of causation -- the proposition, namely, that causes always make their effects more likely, in some appropriate sense.  Is this claim true?  The answer appears to be that it is not, as even some philosophers who are sympathetic to the general idea that there is some connection between causation and probability -- such as Daniel Hausman (1998) -- have realized.  For consider the following.  Assume that there are atoms of type T that satisfy the following conditions:

 (1) Any atom of type T must be in one of the three mutually exclusive states - A, B, or C;

 (2) The probabilities that an atom of type T in states A, B, and C, respectively, will emit an electron are, respectively, 0.9, 0.7, and 0.2

 (3) The probabilities that an atom of type T is in state A is 0.5; in state B, 0.4; and in state C, 0.1.

         Now, given that, for example, putting an atom of type T into state B would be quite an effective means of getting it to emit an electron, it is surely true that, if it is in state B, and emits an electron, then its being in state B is a probabilistic cause of its emitting an electron.  But this would not be so if the above account were correct.  For if D is the property of emitting an electron, the unconditional probability that an atom of type T will emit an electron is given by Prob(D) = Prob(D/A) x Prob(A) + Prob(D/B) x Prob(B) + Prob(D/C) x Prob(C) = (0.9)(0.5) + (0.7)(0.4) + (0.2)(0.1) = 0.75.  But the conditional probability of D given B was specified as 0.7.  We have, therefore, that Prob(D) > Prob(D/B).  So if a cause had to raise the probability of its effect, it would follow that an atom of type T's being in state B could not be a probabilistic cause of its emitting an electron.  This, however, is unacceptable.  So the thesis that a cause must raise the probability of its effect, in the relevant sense, must be rejected.

         The thesis that causes necessarily make their effects more likely is exposed, therefore, to a decisive objection  The basis of this objection is the possibility of there being one or more other causal factors that are incompatible with the given factor, and more efficacious than it.  For, given such a possibility, events of type C may be the cause of events of type E even though the probability of an event of type E, given the occurrence of an event of type C, is less than the unconditional probability of an event of type E.

         But is there nothing, then, in the rather widely shared intuition that causation is related to increase in probability?  The answer is that causation may be related to increase in probability, but not in the way proposed by those who favor a probabilistic analysis of causation.  What this other way is will emerge in section 5.  The crucial point for present purposes, however, is that the relation in question cannot be used as part of a probabilistic analysis of causation, since the relation itself turns out to involve the concept of causation.

 5.4  Conserved Quantities and Continuous Processes

         The three Humean approaches considered so far all offer a reductionist analysis of causation.  Analytical reductionism is not, however, the only form that reductionism with respect to causation can take.  Thus, even if it does turn out to be the case that causal facts are not logically supervenient upon noncausal ones, there is still the possibility of an a posteriori identification of causal and non-causal facts.

         The idea of a non-analytic reduction of causation has been advanced over the past few years by a number of philosophers.  Thus David Fair (1979), for example, proposed that basic causal relations can, as a consequence of our scientific knowledge, be identified with certain physicalistic relations between objects -- relations that can be characterized in terms of the transference of either energy or momentum between the objects involved, while, more recently, Wesley Salmon (1997) and Phil Dowe (2000a and 2000b), have proposed that causal processes are to be identified with continuous processes in which quantities are conserved.  Thus Dowe (2000b, p. 173), for example, suggests the following account:

 Causal Connection:  Interactions I1, I2 are linked by a causal connection in virtue of causal process p only if some conserved quantity is exchanged in I1, and transmitted by p.

         What are the general prospects for a contingent identification of causation with such physicalistic relations?  Perhaps the first point that needs to be made is that once one abandons the view that causal relations are logically supervenient upon noncausal states of affairs, and embraces an a posteriori reduction, one is left with the question of how the concept of causation is to be analyzed.

         But does someone who advances a contingent identity thesis really need to grapple with this issue?  Can it not be left simply as an open question?  Perhaps, but the situation in the case of contingent identity theses concerning the mind suggests that this may very well not be so.  For until a satisfactory analysis has been offered, there is the possibility of an argument to the effect that it is logically impossible for causal relations to be identical with any physicalistic relations.  In particular, might it not plausibly be argued that the concept of causation is the concept of a relation that possesses a certain intrinsic nature, so that causation must be one and the same relation in all possible worlds, just as what it is for something to be a law of nature does not vary from one world to another?  But if this is right, then one can appeal to the possibility of worlds that involve causation, but that do not contain the physicalistic relations in question, or that involve non-continuous causal connections between events, in order to draw the conclusion that causation cannot, even in this world, be identical with the relevant physicalistic relation.

         What is needed, in short, if an a posteriori reduction is to be sustainable, is a satisfactory analysis of the concept of causation according to which causation, rather than having an intrinsic nature, is simply whatever relation happens to play a certain role in a given possible world.  But at present, no such analysis seems to be at hand.

         A second problem for any contingent identification of causation with a physicalistic relation arises from the fact that one needs to find a physicalistic relation that, like causation, has a direction, but where the direction of the physicalistic relation does not itself need to be cashed out in terms of causation.  In Fair's account, for example, the appeal is to the direction of the transference of energy and/or momentum, and this is exposed to the immediate objection that the concept of transference itself involves the idea of causation.

         Fair's response to this problem is that the direction of transference can be explained in temporal terms, rather than causal ones (1979, 240-1).  But this response involves substantial assumptions concerning the relation between the direction of time and the direction of causation.  In particular, many philosophers think that the direction of time is itself to be explained in terms of the direction of causation -- a view that is immediately precluded by Fair's account.

         If, on the other hand, one appeals to features such as the direction of the increase in entropy, or of open forks, etc., to supply the direction for causal processes, one encounters the problem that there are simple worlds, and temporally 'inverted' worlds, that have the same laws, and the same fundamental particles, as our world, but where the contingent identification being proposed either generates the wrong direction for causal processes, or none at all.

        A third difficulty concerns the relation between brain states and the properties of experiences, or between thoughts and decisions and subsequent action.  Thus, many philosophers hold that the phenomenal, qualitative properties of experiences cannot be reduced to non-emergent physicalistic properties.  But if this is right, is it plausible that some quantity is conserved when a brain event gives rise to an experience, or that there is a transference of energy and/or momentum from the fundamental particles of physics to states of affairs involving qualia?  Or is it plausible that when a thought results in behavior, some conserved quantity was transmitted from the thought to the brain?  If these suppositions are not plausible, then any identification of causation with physicalistic relations presupposes the highly controversial claim that the mind involves no properties other than those that are reducible to the properties and relations that enter into theories in physics.

         In view of the above points, the prospects for a physicalistic reduction of causation do not appear bright.