Cause and some Positive Causal Impact

Philosophical Perspectives, 11, Mind, Causation, and World, 1997 CAUSE AND SOME POSITIVE CAUSAL IMPACT Igal Kvart Hebrew University In this paper I put forward a probabilistic analysis of the notion of cause. I argue that for an event A to be a cause of an event C is for A to have some positive causal impact on C. I provide a probabilistic analysis of the notion of some positive causal impact, mainly in terms of the concept of a strict increaser, and argue that, roughly, for A to have some positive causal impact on C is for there to be a strict increaser for A and C. I relate the notion of some positive causal impact to my account of counterfactuals. Finally, I contrast my account with the theories of D. Lewis and W. Salmon. 1. Lewis — Exposition Lewis’s theory of cause is a counterfactual theory.1 Lewis’s conception of counterfactuals (Lewis, 1973) was originally developed for a deterministic world.2 On this account, one orders possible worlds by the relation of intuitive overall similarity, and the counterfactual is (non-vacuously) true iff the consequent is true in all the antecedent-worlds in some sphere (centered around the world in which the counterfactual is assessed, and which includes some antecedentworlds). This theory faced robust counterexamples, in particular counterfactuals which have generally been taken to come out true on Lewis’s theory, though they shouldn’t. One, for instance, is the false counterfactual ‘Had I been at least an inch taller than I am, I would have been exactly an inch taller.’The most notorious counterexample involves the counterfactual ‘Had Nixon pushed the nuclear button, there would have been no nuclear blast’.3 These counterfactuals are patently false, but come out true on Lewis’s theory, one should think, since there are antecedent-worlds in which the consequent is true which are more intuitively similar to the actual world than the antecedent-worlds in which the consequent is false. Lewis consequently modified his original theory by abandoning the reliance on intuitive overall similarity, and moved to a conception of weights and priorities for similarity (Lewis, 1979) 4 on which the overall similarity relation is not 402 / Igal Kvart the intuitive notion.5 It has to be determined by exploring which relation makes the true counterfactuals come out true on this model and the false counterfactuals false. In terms of priorities, Lewis first assigns greater similarity if no major miracles are needed for the requisite changes in the laws than otherwise.6 As the next priority, he assigns greater similarity the more there is perfect match of particular facts. Finally, he assigns greater similarity if no small miracles are involved; he is unwilling to assign weight to similarity of particular facts which fall short of perfect match.7 Lewis’s latest attempt, in his “ Postscripts to “Causation” ” (Lewis, 1986),8 with which we shall be concerned here, is designed for an indeterministic world. On this conception, the only respect of the similarity relation that matters is perfect match of particular facts. When divergences are possible, small miracles are redundant.9 If the antecedent is incompatible with the prior history, backtracking is called for.10 When dealing with causes, however, Lewis stipulates that backtracking is out.11 Let us assume, then, that we deal with cases in which the antecedent is compatible with the prior history.12 On this account, the selected worlds are antecedentworlds where the history prior to the antecedent is the same as in the actual world, and no further constraints are imposed (the laws are those of the actual world). In particular, no constraints are imposed regarding the history from the time to which the antecedent c pertains, tc, onwards.13,14 I later comment briefly on why this conception won’t do. For now, let us look at Lewis’s counterfactual account of causes. 2. Lewis Account of Cause. On this account, c is a cause of e iff c counterfactually raises the probability of e.15 The probability notion here is that of objective chance: the objective chance of e relative to a specific (earlier) time (and thus to the world up to that time). Both Lewis and I resort to objective chances as the requisite notion of probability for the analysis of cause in an indeterministic world.16 But we differ on the time that the chance in question must be relativized to. According to Lewis, “ The actual chance x of e is to be its chance at the time immediately after c” 17 namely, immediately after tc (the time interval to which the cause-candidate c pertains to)—call it tcS . The actual chance of e for Lewis is thus P(e/WcS ). (WcS is the world history until tcS . Note that tc, is, in general, a temporal interval.) 18 The counterfactual chance, on Lewis’s view, is more complex. One considers the most similar antecedent-worlds—those with the same history up to tc as the actual world, Wc, in which ;c is true (and, as usual, assuming indeterminism, with the same laws as the actual world). This is the only constraint placed on the selected worlds. For each such world, Wi, the chance involved is: P(e/WicS ). These conditional chances differ in the probability condition, in particular in the segment of the world history pertaining to the interval tc. The history after tcS doesn’t affect this chance, and that before tc is the same (for each such i). There is thus a Cause and Causal Impact / 403 range of chances, varying with the different worlds. Lewis’s counterfactual probability increase condition is that the actual chance be ‘much higher’ than these hypothetical probabilities.19 This counterfactual probability increase condition is, for Lewis, a sufficient condition for c’s being a cause of e. To turn it into a necessary condition, it must be closed, for Lewis, under cause transitivity (so that if c is a cause of d and d of e, then c is a cause of e).20 Thus, for Lewis, c is a cause of e iff there is a chain of events beginning with c and ending with e where each event in the chain (save the last) fulfils the counterfactual probability increase condition vis-a-vis the next. On this account, then, the counterfactual sufficient condition for c’s being a cause of e is entirely independent of the actual intermediate history between tcS and e. If c and e satisfy the counterfactual probability increase condition, c is a cause of e, regardless of what this intermediate history is. This, I believe, is radically wrong, as I will argue later. Further, Lewis’s account relies on cause transitivity. The view that cause is transitive is common to many past as well as current positions regarding the notion of cause, in particular in the tradition of probabilistic cause. I will argue, however, that cause transitivity does not hold, undermining Lewis’s and other accounts. 3. Salmon — Exposition In his book Scientific Explanation and the Causal Structure of the World,21 Salmon actually put forward two distinct theories. On the first, which he endorsed, the basic notion is that of a process—indeed, he seeks to replace event ontology with process ontology. Processes for him include material objects over time (e.g., this chair) as well as light rays, etc. His main concern is the distinction between causal and noncausal processes. An example of a noncausal process would be a shadow cast by a moving object. Pseudo (i.e., noncausal) processes are processes that need not adhere to the principle of the special theory of relativity involving causal signals, i.e., the principle that they can’t travel faster than light. Salmon’s criterion for a causal process was formulated in terms of mark transmission (MT).22 Basically, the criterion is as follows: a causal process is one that can transmit a mark. A mark is transmitted only when two processes intersect in a causal interaction (to be defined later). A mark is transmitted by a process P after a single local interaction with another process when there is a modification Q9 of a feature Q of P, and this modification Q9 persists over a certain time interval without further interventions.23 The last qualifying clause itself invites a counterfactual interpretation. But Salmon imposes a further heavy counterfactual constraint. The process P must be such that, in the absence of any interactions with other processes, it would manifest the original feature Q over a certain time interval. Salmon defines the notion of a causal interaction (CI) in a manner similar to his MT criterion.24 Two processes, P and P9, undergo a causal interaction in case they intersect and each manifests a modification in a feature and these modifica- 404 / Igal Kvart tions last for a certain time interval. But in addition, a heavy counterfactual qualification is imposed: Were it not for the intersection with the other process, each process would have persisted in exhibiting the original feature in question unmodified over a certain time interval. Salmon is well aware that his definition of a causal process makes substantial use of counterfactual conditions; yet Salmon doesn’t have or endorse a theory of counterfactuals.25 Salmon does not express concern about circularity in his use of counterfactuals for the characterization of causal concepts. His problem with counterfactuals in MT and CI is the common concern that counterfactuals are context-dependent and thus not objective, whereas his distinction between causal and noncausal processes is designed to be objective. He cites Quine’s famous example of the Bizet-Verdi counterfactuals as an instance of context-dependence. Yet he is convinced that counterfactuals are indispensable for science and that the type of counterfactuals utilized in MT and CI is objective. He then illustrates this with experiments designed to justify the conclusion that the pertinent counterfactuals are true. Such experiments establish not only their truth but also their objective character. Any adequate analysis of counterfactuals, on Salmon’s view, should yield these results. In my view, Salmon is right that some types of counterfactuals are objective and some aren’t (though he is wrong in accepting Quine’s example as illustrating context-dependence 26 ). As I have argued at length, the basic types of counterfactuals are indeed generally objective.27 However, though he hopes to solve Hume’s problem of causation on Hume’s terms, Salmon is on shaky ground regarding whether or not an adequate analysis of counterfactuals can be Humean.28 The main question Salmon does not face is whether invoking counterfactuals results in circularity that undermines his program of analyzing causal notions. Unfortunately, as we will see below (section 13), the counterfactual construction involves the notion of cause or closely related notions, and therefore the analysis of causal notions in terms of counterfactuals is circular. Using the notions of causal process and causal interaction, Salmon sketches a notion of a non-simultaneous cause as follows. c is a cause of e if there is a causal process P, such that a causal process P9 causally interacts with it at c, and another causal process P0 causally interacts with P at (the later) e.29 This notion of cause, in its present form, is inadequate. Any two events (involving two interactions) connected by some body (which is a causal process) equally exemplify this would-be cause-effect relation.30 Suppose someone comes to a gate, finds it locked (the office is closed), kicks it in frustration and leaves. Ten minutes later someone else (unaware of the first) comes to the gate and also kicks it in frustration. These two events are two causal interactions with the same body, and thus fulfill Salmon’s paradigm. But surely the first event is no cause of the second.31 Salmon’s process-cum-interactions theory of cause-effect is thus as yet just a programmatic suggestion which does not deal with the fundamental difficulties involved in analyzing the notion of cause. Cause and Causal Impact / 405 This shortcoming has nothing to do with the failure of cause transitivity, which I discuss later, nor does it depend on the adequacy of Salmon’s characterizations of causal interaction (in CI), mark transmission (in MT) and causal processes. Regarding events involving the same process (say, the same object), surely some are cause-effect pairs and some are not. Salmon’s account and his discussion do not tell us which are and which are not.32 The problem of telling which pairs of events are cause-effects pairs and which are not applies to pairs separated by numerous pertinent causal interactions, as well as to pairs which consist not of interactions involving the same process but of those involving distinct processes; 33 Salmon’s account cannot determine which are which. Further, cause-effects pairs which involve no connecting process (e.g., omissions) threaten to undermine Salmon’s very paradigm. So Salmon’s account and his basic concepts of causal process and causal interaction do not suffice to yield an adequate analysis of what a cause is.34 4. Salmon’s Second Theory This theory is a version of the probability increase doctrine. On this doctrine, the leading idea is that a cause increases the probability of the effect. However, a cause nonetheless can lower the probability of the effect. To overcome this problem, the idea of interpolation was put forward—there must be an intermediate event d such that c • d raises the probability of e.35 However, here too counterexamples have been devised: cases where it seems that even with an intermediate event, or a series of intermediate events, c does not increase the probability of e (e.g., the golf ball and branch case).36 The sense that interpolation doesn’t yield an adequate account is indeed well-founded, although such counterexamples do not seem to bring out the primary reason for this. Take some non-cause of e, c, which lowers the probability of e. Then select d, which is a cause of e, and which, together with c, raises the probability of e. On this account, the non-cause c comes out a cause. (Suppose, in a suitable version of the golf example, that c is: the ball is brown. Then take f as: the ball is close to and moving toward the hole. c • f raises the probability of e, but c is no cause of e, though f is.) Thus, choosing a cause d can yield that c • d increases the probability of e even if c is not a cause. Instead of adding such intermediate events to the initial probability condition, Salmon’s own suggestion employs what he calls successive reconditionalization. That is: there must be a series of intermediate events such that c raises the probability of the first, the first raises that of the second, and so on until the last intermediate event raises the probability of e. Each stage, then, involves conditionalizing on the previous stage only. Here Salmon obviously relies on cause transitivity.37 Salmon’s second theory, then, like Lewis’s theory, hinges on cause transitivity, to which we now turn. (The reader should bear in mind, though, that, unlike Lewis’s account, events, as employed in this paper, are narrowly individuated.) 38 406 / Igal Kvart 5. Cause Transitivity Cause transitivity is widely accepted as a cornerstone of the analysis of cause. It is, however, invalid. Consider the following example.39 A worker x was injured in a work-related injury. Let c be: x’s finger was cut off at t1 by a machine. His co-workers rushed him and the dismembered digit to a hospital, where an expert surgeon was on alert; let d be: the surgeon reconnected the finger to x’s hand at t2. The surgery was a great success, and indeed, e: x’s finger was functional at t3. c, d and e occurred at t1, t2 and t3 respectively. Clearly, intuitively, c is not a cause of e—the fact that x’s finger was cut off at t1 is not a cause of the fact that x’s finger was functional at t3. But, again, intuitively, c is a cause of d—the fact that x’s finger was cut off at t1 is a cause of the fact that the surgeon reconnected the finger at t2; and, again, intuitively, d is a cause of e—the fact that the surgeon reconnected the finger at t2 was a cause of the fact that x’s finger was functional at t3. Hence cause transitivity fails. This is not an isolated example. In fact, it covers a whole range of similar cases. Lewis’s theory and Salmon’s second theory explicitly require, indeed, hinge, on cause transitivity, and thus fail if cause transitivity is invalid. Even Salmon’s first theory would seem to presuppose cause transitivity in fact. Thus, suppose P9 causally interacts with P, after which P causally interacts with P99, and then P99 causally interacts with P999. Assume, with Salmon, that the first interaction was a cause of the second, which was a cause of the third. Was the first a cause of the third? Salmon would presumably say yes (at least in standard cases of this sort). But Salmon has no way of accounting for the cases in which the first is indeed a cause of the third if, by hypothesis, these two interactions do not have a shared process. Such cases fall outside the model Salmon puts forward. Assuming cause transitivity will help with many such cases, but of course, will not adequately handle cases characterized by failure of cause transitivity. Salmon doesn’t explicitly notes the need for cause transitivity on this first theory of his; he presumably would be amenable to it here, since he is with regard to his second theory. But it would yield an additional source of failure for this account, since here, as in his second theory, and in general, cause transitivity is not valid. 6. Counterfactuals On the account of counterfactuals I developed, a central class of counterfactuals is that of counterfactuals A.B whose antecedent A and prior history (i.e., Cause and Causal Impact / 407 the history of the world up to tA) are compatible.40 I shall confine my discussion here to these counterfactuals (so-called n-d counterfactuals). Their truth-condition amounts to the following: a counterfactual A.B (of this sort) is true in case the consequent B can be inferred from the antecedent A plus implicit premises. These implicit premises include the history of the world prior to A, as well as the laws of nature. In addition, two crucial sets of true factual statements pertaining to the temporal interval between A and B must be included. The first is the set of true factual statements pertaining to (tA, tBO ) 41 to which A is causally irrelevant.42 The second is the set of true factual statements pertaining to (tA, tBO ) to which A is purely positively causally relevant. Thus, the counterfactual schema is as follows: (I) A.B is true iff A, together with WA and the set of true statements in WA,BO 43 to which A is causally irrelevant or purely positively causally relevant,44 implies B (via the laws).45 (An actual event F may have a mixed causal impact on a (later) event E: partly positive, partly negative. This situation is quite common: often times, the positive causal impact is transmitted through one route, the negative causal impact through another. If F has some positive causal impact on E but no negative causal impact at all, then F has purely positive causal impact on E.46 ) To illustrate the importance of these two sets of statements in the inferential schema (I), consider the following examples.47 The first illustrates causal irrelevance.48 Someone (x) contemplated selling certain stocks. After deliberating, x (at t1) instructed his agent to sell (A).49 A couple of days later (t2) the stock market skyrocketed (E). At t3 50 it would be true to say: Had x not sold my stock, he would have been rich today. This counterfactual is intuitively true. But to come out true on the inferential model, the event E—that the stock market skyrocketed at t2— must be retained among the true statements from which B is to be inferable (through the laws—see schema (I)). And indeed, A is causally irrelevant to E. Thus, just A and its prior history WA do not suffice to yield B (x is rich today) through the laws. A different intermediate history WA,BO that does not include events such as E, would not support this counterfactual. The truth of A.B thus hinges on whatever transpired between A and B, in particular, on events such as E. Hence the truthconditions of A.C must be a function of WA,BO . An analysis which is not a function of WA,BO therefore will not do (e.g., Lewis’s Analysis 1 in “Counterfactual Dependence and Time’s Arrow”, which is not a function of the intermediate history WA,O BO 51 ). A second example will highlight the importance of purely positive causal relevance. An architect (x) participated in a highly competitive design competition. Participants submitted architectural plans and a budget, including the architect’s fee. x won the contest (E), and was awarded the contract to design the building. He now said: Had I requested $1000 less for my fee (than I actually requested), a fortiori I would have been awarded the contract. This counterfactual is true, but will not come out true on the inferential model unless E is retained 408 / Igal Kvart in the inferential schema. The antecedent of this counterfactual is indeed purely positively causally relevant to E. It is thus clear that these two sets of statements—the ones consisting of true statements in WA,BO to which A is causally irrelevant, and the other to which A is purely positively causally relevant—must be retained when a counterfactual A.B is assessed. Counterfactuals with true consequents are called semifactuals. A corollary of the above analysis is that semifactuals whose antecedents are causally irrelevant to their consequents are all true (since the consequent is retained among the implicit premises), and similarly semifactuals whose antecedents are purely positively causally relevant to their consequents are true as well, for the same reason.52 Lewis ignores the events in the (tcS , teS ) interval, preserving only the history prior to c and the laws. As a result, his theory cannot handle, in general, cases that rely on the preservation of these sets of statements describing intermediate events. Of course, for Lewis it would be conceptually difficult to accommodate these statements since, in the first place, he has no systematic conception of causal relevance. Perhaps more seriously, causal relevance and purely positive causal relevance are causal notions, and Lewis is committed to reducing causes to counterfactuals, while this approach seems to call for reductive efforts in the other direction.53 The notion of purely positive causal relevance (or purely positive causal impact—I use the terms here interchangeably 54 ) has cognates. Consider a counterfactual ;A.C (A being actual). True factual statements in (tA, tCO ) to which the antecedent, ;A, is purely positively causally relevant are preserved. ;A is purely positively causally relevant to an intermediate statement D just in case the actual A is purely negatively causally relevant to D, that is, just in case A confers purely negative causal impact on D.55 Not to confer purely negative causal impact on D is either to be causally irrelevant to D, or else to confer some positive causal impact on D. The notion of some positive causal impact therefore resurfaces in straightforward reformulations of the notions of purely positive causal impact and causal relevance. This notion is, on my account, of the highest importance. My main thesis in this paper can now be stated: Being a cause is having some positive causal impact. That is: Thesis 1: c is a cause of e iff c has some positive causal impact on e. I shall now turn to the probabilistic analysis of the notion of some positive causal impact. 7: Some Positive Causal Relevance We may, at this point, be reminded that the notion of a cause, as discussed and analyzed in this paper (and similarly, of course, the notion of some positive Cause and Causal Impact / 409 causal impact), is considered strictly on the token level, namely, as relating particular event tokens. (The corresponding generic relations are not discussed here.) It is to be clearly distinguished from the very different notion of the cause, which is context-dependent and interest-relative, unlike, as we will see, the notion of a cause, which turns out to be objective. My analysis is based on the notion of objective chance-like probabilities.56 The basic notion is that of P(C/Wt)—the chance of an event (specified by C) given the state of the world at t (t,tc). (The notion of chance, as I use it, is not very different from Lewis’s.) 57 Under what conditions does an event A have some positive causal impact on C? Surely, for A to have some positive causal impact on C, A must be causally relevant to C; so we shall assume that A is causally relevant to C.58 The condition that next comes to mind for securing the relation of some positive causal impact (henceforth: spci) can be called the ab initio probability increase condition, or aipi: (1) aipi: P(C/A • WA) . P(C/;A • WA) However, this condition is neither sufficient nor necessary for A’s having de facto spci on C, because aipi does not depend on WA, C, the world history between A and C,59 and thus only secures expected positive causal impact of A on C given WA.60 But expected positive causal impact need not coincide with some de facto positive causal impact: the latter does indeed depend, quite strongly, on the intermediate history WA, C. Thus, suppose aipi holds. Suppose, further, that there is an actual intermediate event E between A and C (i.e., in (tA, tC) 61 that reverses the probability increase of aipi, thereby eliminating the indication of spci; that is, an event that reverses the positive probabilistic impact reflected in aipi when added to the condition on both sides. Now an event E that preserves the probability increase relation of aipi fulfills: (2) P(C/E • A • WA) . P(C/E • ;A • WA). However, an (actual) intermediate event reverses the indication of spci provided by (1) by fulfilling: (3) P(C/E • A • WA) , P(C/E • ;A • WA). In this type of case we will call E a decreaser for A and C: once added to the condition on both sides, it reverses the probability increase condition aipi, generating the probabilistic decrease condition (3). Similarly, ab initio probability decrease, or aipd, i.e.: (4) P(C/A • WA) , P(C/;A • WA) (which is the mirror image of aipi), does not capture the notion of some negative causal impact, since rather than reflecting some de facto negative causal impact, 410 / Igal Kvart it captures only the relation of some expected (as of tA 62 ) negative causal impact. Again mirroring the case of aipi, an (actual) intermediate event E reverses the probability decrease of aipd, and thus the indication of some negative causal impact, if it yields: (5) P(C/E • A • WA) . P(C/E • ;A • WA). Such an event E will be called an increaser (for A and C). Both increasers and decreasers reverse the initial probabilistic indication (as of tA), and are therefore called reversers. (A reverser, then, is either a decreaser or an increaser.) 8: The Dictator-Rebels Example An example will help establish that aipi and aipd do not reflect the requisite notions of some positive and some negative causal relevance. Example 1. Consider a particular country in which the US has a military base. The chief US policy consideration regarding that country is to secure the long-term presence of this base, which requires the local government’s consent. At t0, when the story begins, the country is ruled by a military dictator who supports the presence of the US military base but expects US support for his regime in return. There is also a rebellious opposition movement, but it is relatively weak and has no serious chance of overthrowing the regime. At t0, the military dictator requests arms from the US to help suppress these rebels. At t0, for the US not to approve this request is viewed in Washington, correctly, as seriously decreasing the prospects for continued American presence at the base. Providing the arms is viewed, again correctly, as a move that would greatly alienate the rebels. At t1 (some time after t0) the US decides to provide the requested arms. Call this event A. However, the improbable (as of tA) happens, and after some struggle, the dictator is deposed. The rebels seize power shortly thereafter (at t2; call this event E), and order the US to leave the country. The US acquiesces, losing the base (at t3; call this event C). Ex post facto, as things turned out,63 A clearly had spci on C. However, at time tA the probability of the rebels’ coming to power was very low, and thus: (4) P(C/A • WA) , P(C/;A • WA). That is, at tA the chance that the US would no longer have the base (at t3) was higher were it to abstain from supplying the requested military aid to the dictator (;A) than were it to comply. Thus, aipd (i.e., (4)) obtains. Hence aipd is compatible with spci (and does not capture the relation of some negative causal relevance—henceforth: snci 64 ). Intuitively, however, A is clearly a cause of C: that the US decided to provide the requested arms was a cause of the fact that it lost the base. This is our first illustration that spci and cause coincide. To illustrate the corresponding failure of aipi to capture spci, consider a mirror image of this example. Imagine the story modified as follows: The US Cause and Causal Impact / 411 in fact refuses at t1, out of moral considerations,65 to supply the requested arms (call this A9 ).66 The dictator prepares to terminate operation of the US base, but, as in the original version, is deposed, and the rebels take over (E). In spite of their appreciation of the American refusal to arm the dictator, the rebels decide to assert their sovereignty, and revoke permission for continued American operation of the base. Consequently, as in the original version, the US abandons the base at t3 (C). Clearly the fact that the US refused to send arms (the actual A9 ) had snci on C. In particular, intuitively, A9 is not a cause of C. Furthermore, A9 does not have some positive causal impact on C at all: it is purely negatively causally relevant to C. However, the story and its probabilistic underpinnings are the same in the two variations (up to tA, which is the same as tA9). Thus, at tA9, the fact that the US refused to supply arms to the dictator increased the probability of its having to abandon the military base, yielding aipi: (19) P(C/A9 • WA9) . P(C/;A9 • WA9). Hence, aipi is compatible with snci. As this example shows, WA, the history of the world up to tA, does not fully capture the de facto relations of spci (and snci) between A and (the later) C. On the conception which underlies my approach, causal relations are determined by the pertinent facts and the probabilistic structure of the world. But it is the factual (and probabilistic) structure of the world history up to tC (and not just up to tA) that determines the causal relations between A and C. Condition (1) reflects only WA and the probabilistic structure given WA; it thus does not adequately reflect the pertinent constitution of the world in its entirety. In particular, it does not reflect WA,C, and thus is not equipped to capture the de facto causal relations between A and C, which are a function of WA,C. 9: aipi, spci, aipd and snci aipi, then, is not a sufficient condition for spci of A on C, since a decreaser can counter it. However, if aipi obtains and there is no decreaser, that is, if aipi is unreversed, then aipi conclusively establishes the presence of spci of A on C. This typically happens in straightforward cases (i.e., those with no probabilistic turbulence in (tA, tC)), which are the norm. Consider such a case: Example 2. An individual, a, instructs her stockbroker to sell her stocks (call this A),67 and the latter proceeds to do so. Take as C: a’s stocks are sold. In a normal case of this sort, A has spci on C, and indeed: (1) P(C/A • WA) . P(C/;A • WA).68 Hence the case is one of aipi, and there is no reverser E (fulfilling (3)). As a case with unreversed aipi, there is spci of A on C. And intuitively, A is clearly a cause of C, again illustrating that being a cause is a case of spci. 412 / Igal Kvart (Consider the case of aipd, i.e.: (4) P(C/A • WA) , P(C/;A • WA). In analogy with the reasoning above, if aipd is unreversed, i.e., if there is no increaser—that is, no event E in (tA, tC) fulfilling (5)—then the case is one of snci (of A on C). If there is a reverser E, however, it need not be a case of snci.) 10: Strict Reversers An aipd case with an increaser E thus may well be a case of spci. In fact, it will be a case of spci if the increaser E is not further reversed. Similarly, an aipi case with a decreaser (as in (3)) which is unreversed is a case of snci. Thus, in a case of aipi with a decreaser E there could be an event E1 that restores probability increase if added to the conditions on both sides of (3), thereby reversing the probability decrease of the decreaser E, and restoring apparent spci. That is, there could be an actual event E1 (in (tA, tC)) such that: (7) P(C/E • E1 • A • WA) . P(C/E • E1 • ;A • WA). In case of a decreaser E with such an E1, an increaser for it, we shall say that E has been reversed. A case with an increaser for a given decreaser such as (7) thus restores apparent spci. An increaser which is not reversed (i.e., with no decreaser for it) will be called a strict increaser. (Similarly, an unreversed decreaser, i.e., one with no increaser for it, will be called a strict decreaser). Accordingly, an unreversed reverser will be called a strict reverser. A strict increaser conclusively establishes spci, and a strict decreaser conclusively establishes snci. In example 2 above, E is unreversed: there is no actual E1 fulfilling (7) for E, and the case is clearly one of spci. We can thus formulate a primary tenet of our argument as follows: Thesis 2:A strict increaser establishes spci. Consequently, in view of thesis 1 above: Thesis 3:A strict increaser establishes a cause. A case of aipi can be considered a degenerate case of an increaser,69 and a case of aipi without a decreaser can be considered a degenerate case of a strict increaser. We shall thus consider such cases of aipi as having a null increaser and a strict null increaser respectively. (The analogous cases, for aipd, will be considered as having null decreasers and strict null decreasers).70 Cause and Causal Impact / 413 Thus far we considered cases of aipi and aipd. We now consider the remaining cases, those characterized by equi-probability (EP). A case of EP is one in which: (8) EP: P(C/A • WA) 5 P(C/;A • WA). A case fulfilling EP may or may not be a case of spci, depending on whether there is an increaser (an event fulfilling (2)), and in particular, a strict increaser. A strict increaser will yield spci in an EP case, as in an aipd case. A strict decreaser will, similarly, yield a case of snci. We are thus in a position to further generalize as follows: Thesis 4: To be a cause is to have a strict increaser. That is: A is a cause of C iff there is a strict increaser for A and C (and A is causally relevant to C).71 11: More on Strict Reversers If an aipi case has no reverser, the aipi condition (1) itself establishes a case of a degenerate, or null, strict increaser. (Regardless of whether or not there is a reverser, (1) establishes a null increaser). Similarly, (4), the aipd condition, establishes a case of a null decreaser, which is a null strict decreaser if it has no increaser. Anull strict increaser, like a non-null strict increaser, establishes a case of spci (and a null strict decreaser, like a non-null strict decreaser, a case of snci). Null increasers and null decreasers will be considered special cases of increasers and decreasers. The general result, argued for above, together with the present observation, a special case of it, can be summed up by saying that a case with a strict increaser— null or not—is one of spci, whereas a case with a strict decreaser is one of snci. In general, however, an aipi case can have a decreaser E1 which has a further reverser E2, i.e.: (9) P(C/A • E1 • E2 • WA) . P(C/;A • E1 • E2 • WA). E2 will thus be considered an increaser for E1 (relative to A and C). E1 • E2 can still have yet another reverser (that is, a decreaser for E1 • E2) E3, i.e.: (10) P(C/A • E1 • E2 • E3 • WA) , P(C/;A • E1 • E2 • E3 • WA). And so on: E1 • E2 • E3 can have a further increaser, etc. E2’s being an increaser for E1 implies that E1 is not a strict decreaser; and E3’s being a decreaser for E1 • E2 implies that E1 • E2 is not a strict increaser. An increaser E for A and C (that is, when (5) is satisfied, regardless of any other intermediate event E9 ) will be an increaser simpliciter; and likewise for decreasers. What counts for spci and snci in the results above are strict increasers and strict decreasers (which are always increasers or decreasers simpliciter). The importance of strict increasers is thus clear: A strict increaser for A and C yields the result that A is a cause of C. Ex post facto probability increase is the 414 / Igal Kvart core import of being a cause, which is in turn captured by the presence of a strict increaser. While aipi and aipd are not conclusive indicators of ex post facto probability increase or decrease, the presence of a strict increaser does conclusively establish ex post facto probability increase, the hallmark of spci, which in turn constitutes being a cause. Thus, an aipd case with a strict increaser is a case of spci. Similarly, an aipi case without a decreaser, i.e., a case with a null strict increaser, is also one of spci. One case, however, has not yet been taken care of. Suppose A has aipi on C, but there is a decreaser E. However, suppose that E can be reversed by E1 (which is thus an increaser for E), but that E1 on its own (apart from E) preserves the probability increase of aipi, and that E1 cannot be further reversed. That is, in addition to aipi, we have (11) P(C/A • E • WA) , P(C/;A • E • WA), but we also have: (12) P(C/A • E • E1 • WA) . P(C/;A • E • E1 • WA), and even further: (13) P(C/A • E1 • WA) . P(C/;A • E1 • WA), in a non-reversed way, i.e., for no E2: (14) P(C/A • E1 • E2 • WA) , P(C/;A • E1 • E2 • WA). In line with our reasoning till now, this constitutes a case of ex post facto probability increase, and thus establishes the case as one of spci. However, in our terminology, E1 is not an increaser, since the case is already one of aipi, and we defined increasers simpliciter only for cases of aipd. Let us therefore extend our terminology so as to allow for E1 to qualify as a strict increaser, as follows: Strict Increaser Extended Definition: If A and C yield a case of aipi, and the null event is not a strict null increaser, and E1 is an intermediate event (i.e., in (tA, tC)) which, when added to the condition on both sides, yields a probability increase which is not in turn reversed (i.e, it fulfils (13), and also (14)—for no E2), E1 will be considered a strict increaser for A and C as well.72,73 Given this extension of the notion of a strict increaser, the import of our theses is broadened, allowing cases such as the one just examined to qualify as cases of spci, as indeed they should. The thesis that spci consists in there being a strict increaser thus becomes richer and more accurate .74,75 Cause and Causal Impact / 415 In general, even in cases where there is a strict increaser or a strict decreaser, there may be other increasers or decreasers (simpliciter), and there may be chains of events ^E1, ... , En& where Ei is a decreaser for E1 • ... • Ei21 when i is, say, even, and an increaser when i is odd (1,i#n). The presence of such chains has no bearing on whether the case is one of spci or snci beyond some possible bearing on whether there is a strict decreaser or a strict increaser: it is the latter that makes the case one of spci or of snci. Thus, without a strict increaser (null or not), a case will not be one of spci, and without a strict decreaser (null or not), it will not be one of snci. A case with a strict increaser as well as a strict decreaser will be one of both spci and snci; in particular, the cause-candidate will qualify as a cause. The upshot of our analysis, then, is: Thesis 5: A has some positive causal impact on C iff there is a strict increaser for A and C.76 (Of course, under our ongoing assumption of the causal relevance of A to C.) More precisely, A has spci on C iff A is causally relevant to C and there is a strict increaser for A and C. Previously, we argued that A is a cause of C iff A has spci on C. Combined with the present claim, it yields: Thesis 6: A is a cause of C iff A has spci on C iff there is a strict increaser for A and C. 12: The Electric Circuit Example. I now consider a challenging case for the above analysis. It is a variation on a general type of case in which A, the cause, initiates a course of events with a low probability of leading to the effect C, rather than a course with a high probability of leading to C.77 In the specific example we will consider, there is a wire leading to a fork-type switch governing a divergence into two routes, each leading to a bulb. In one position, the switch channels the current to a lower route, and the probability that the current so channelled will go all the way through to the bulb is only 0.5. In fact, the switch was in this position, that is, A occurred. The other route is very reliable: when the switch is in this other position (i.e., ;A—this is the only other option), it channels the current to the upper route, and the probability of the current’s going through to the bulb is 0.9. Thus: ~A —————– y ——– switch bulb A | | The bulb indeed lighted up (C). Intuitively, A is a cause of C. 416 / Igal Kvart The challenge in this case is to find an increaser. The obvious pool of candidates seems to yield none: none of the events involved in the actual route (the lower course) yields an increaser. However, consider events along the untraversed route, and in particular actual events featuring the non-occurrence of nonactual events that would have occurred had this route been traversed. Thus consider: E—there is no current on the upper branch at point y during (tA, tC). (y is very close to the bulb). Surely A yields aipd: P(C/A • WA) , P(C/;A • WA). But given E, A increases the probability of C: on the right-hand side, with ;A, we have the probability of C given that the switch is turned toward the upper route (only) with, however, no current going through y during the (tA, tC) interval—a probability close to 0. On the left side, we have the probability of C given A • E, that is, given the switch’s opening to the lower channel with no current going through the upper channel— medium probability, the same as without E. Hence E is an increaser, in fact, a strict increaser. E thus signifies the non-occurrence of a hypothetical event that would in all likelihood have taken place had the upper route been selected. It illustrates a general type of increasers—the non-occurrence of a hypothetical event in the non-traversed route. 13: The spci Conception of Cause and the Analysis of n-d Counterfactuals On the account of counterfactuals I presented above (the n-d type), the statements to be preserved in assessing a counterfactual ;A.B (A being true) are the true factual statements pertaining to (tA, tBO ) specifying events to which the antecedent-event ;A is either causally irrelevant or else purely positively causally relevant.78 Hence they are the true statements specifying events to which the actual A is causally irrelevant or purely negatively casually relevant.79 That is, they are those statements specifying events to which A is either causally irrelevant or else causally relevant but without any positive causal relevance. Put differently, they are those statements specifying events to which A is either causally irrelevant or else causally relevant but not a cause of. If A is causally irrelevant to E, A is surely not a cause of E. The set of actual events to which A is causally irrelevant or else causally relevant but not a cause of is thus exactly the set of actual events in (tA, tBO ) of which A is not a cause. (The remaining set of events in that interval is the set of intermediate events on which A has some positive causal impact—i.e., those of which A is a cause.) We can thus reformulate the account of n-d counterfactuals thus (for a true A): (II) ;A.B is true iff B is inferable (through the laws) from ;A, WA and the set of true factual statements in (tA, tBO ) of which A is not a cause A.80 The analysis of counterfactuals I have presented thus looks as if it relies on the notion of cause. I have, however, provided an independent probabilistic anal- Cause and Causal Impact / 417 ysis for the notion of cause through the probabilistic accounts of causal irrelevance and spci, and thus for the notion of cause that figures in the above analysis. However, my analysis bodes ill for certain attempts to reduce cause to counterfactuals. Apart from Lewis’s account with its special features, construed as aiming at reducing causes to counterfactuals, there is a more general line of thinking, going back to Hume, that takes the counterfactual ;c.;e to be the key to the analysis of being a cause, along with some needed means of characterizing (counterfactually or otherwise) 81 cases of so called overdetermination and of distinguishing actual from merely potential causes in such cases. However, this approach seems to be undermined in view of the account above, since, as we have seen, the analysis of counterfactuals requires a notion akin to that of being a cause, thereby leading to circularity.82 14: Counterfactuals and Causes. I will now relate the above account of counterfactuals to this venerable counterfactual doctrine of cause, i.e., the theory that ;A.;C captures the notion of cause (possibly, in conjunction with closure under transitivity). (Lewis upholds this conception for the case of a deterministic world, but not for the indeterministic case, where he upholds an account based on counterfactual conditional chances, a position we discussed above (section 2) and will assess below (sections 15 and 16).) I have now argued that because counterfactuals invoke causes, to analyze cause in terms of counterfactuals is, in general, circular. But I will ignore this problem here, and concentrate on the extensional adequacy of this doctrine. Examination of counterfactuals in which the consequent is underdetermined by the antecedent plus the implicit premises 83 reveals that this strategy is not promising. Discussion of the law of counterfactual excluded middle made clear that in many cases neither ;A.;C nor ;A.C is true.84 Yet in many such cases, A qualifies as a cause of C: the counterfactual condition ;A.;C thus fails as a necessary condition 85 (even with closure under transitivity). Consider an example: Example 3. Three men, #1,#2 and #3, attacked a fourth, u, overcame him and dragged him to a river. All the three took part, to about the same degree, in trying to keep him submerged until he drowned. Consider: A—the 3 attackers endeavored to keep u’s head submerged. and in particular: Ai —attacker #i endeavored to keep u’s head submerged. And indeed: C—u drowned. Surely, intuitively, each of the Ai’s was a cause of C. 418 / Igal Kvart In particular, A1 (the fact that #1 endeavored to keep u’s head submerged) was a cause of C (the fact that u drowned). Yet in this case, the counterfactual ;A1.;C, namely, if attacker #1 had not endeavored to keep u’s head submerged, u wouldn’t have drowned, is false.86 In cases like this, where the relevant counterfactual is false, invoking cause transitivity seems of no avail. Hence the conception of the counterfactual condition as an elucidation of being a cause (even in cases not commonly classified as cases of overdetermination, such as this case 87 ) fails. This is typical in cases with a number of contributing causes.88 The counterfactual condition in such cases requires that, without the event A, the outcome C will not have take place. Such a sine qua non condition (in cases where cause transitivity is of no avail) is too strong to capture the notion of being a cause: 89 the counterfactual (even with cause transitivity) is too strong to constitute a necessary condition for being a cause (even when cases of overdetermination are excluded). Yet in such cases, the event in question, which is indeed a cause, clearly has some positive causal impact on the effect: in our case, A1 (the fact that attacker #1 endeavored to keep u’s head submerged) has spci on C (the fact that u drowned). An expectation that the notion of cause can be characterized by the counterfactual condition ;A.;C (along with cause transitivity) may be taken to indicate an expectation that being a cause involves positive causal impact that is crucial for the occurrence of the effect. Despite this, in many cases causes merely have some positive causal impact on the effect.90 Yet spci doesn’t suffice to ensure the truth of the counterfactual ;A.;C, which is the heart of the counterfactual condition’s failure to be necessary for being a cause. The falsehood of the counterfactual ;A.;C is entirely consistent with A’s being a cause of C.91 Yet the intuitive appeal of the association of cause with the counterfactual condition ;A.;C is sufficiently strong that an explanation of the relationship between this counterfactual and the notion of spci is called for, especially in conjunction with the account of counterfactuals I outlined above. The core of this connection is that the counterfactual ;A.;C is indeed a sufficient condition for A’s being a cause of C. Let me elaborate. Suppose ;A.;C is true (A, C are actual). Then ;A.C is false (the truth of ;A.C is inconsistent with the truth of ;A.;C). In particular, A is causally relevant to C (otherwise ;A.C is true). Further, it is false that ;A has purely positive causal impact on C (which would be sufficient for the truth of ;A.C). Hence it is false that A has purely negative causal impact on C.92 Since A is causally relevant to C, yet doesn’t have purely negative causal impact on C, it follows that A has some positive casual impact on C. Hence A is a cause of C. Thus, on my account of counterfactuals, together with the thesis offered here, namely, that to be a cause is to have spci, it follows that if ;A.;C is true, then A is a cause of C, i.e., that the counterfactual conditional is indeed a sufficient condition for being a cause. This venerable intuition, a cornerstone of the counterfactual conception of cause, is quite valid. That this account of counterfactuals, and in particular the role of spci,93 together with the thesis that cause Cause and Causal Impact / 419 consists in spci, yields, as a logical consequence, that the truth of the counterfactual is a sufficient condition for being a cause, constitutes, as I see it, a vindication of these two conceptions, as well as a vindication of the affinity between the counterfactual construction and the notion of cause. 15: Lewis and Salmon Revisited: Failure Due to Strict Decreasers Lewis’s account of causation compares the actual chance P(e/WcS ) with the range of chances in non-actual possible worlds P(e/WicS ) (where Wi is a non-actual possible world). The main sufficient condition for c’s being a cause of e is that the first probability be much greater than the range covered by the latter. Consequently, this analysis leaves out altogether two major categories of intermediate events. The first is that of causal breakers—events that truncate the would-be causal chain from c to e, in cases where c is not a cause of e since c is not causally relevant to e (i.e., e is causally independent of c). Yet even when there are such causal breakers, the above sufficient condition may well obtain.94 The second category of events his theory ignores is that of increasers and decreasers, i.e., reversers. We saw the crucial role that reversers play regarding whether c is a cause of e. But Lewis’s account accords reversers no role whatsoever. As a result, the above sufficient condition may obtain even though there is a decreaser that would reverse the probability increase and (in the absence of a strict increaser) render c a non-cause of e, as, for instance, in the example of the dictator considered in section 8. In insisting that the sufficient condition for being a cause is independent of the intermediate history between tcS and e, Lewis is oblivious to the diverse ways in which intermediate events (and probabilistic features thereof ) can foil or establish a cause regardless of the indications generated by probability increase or probability decrease conditions, however conceived.95 Indeed, on my view, the role of such intermediate events constitutes a central aspect of the probabilistic picture of being a cause, and should thus be a key component of probabilistic analyses of cause. To see how its failure to attend to decreasers impacts on the adequacy of Lewis’s analysis, consider the variation on the dictator example from section 8 used to demonstrate that a case of aipi need not signify the presence of a cause. Modify the original version by supposing that, contrary to prevailing geopolitical wisdom, the US refused to send arms to the dictator—A9 (A9 is close to ;A in the original variation; the two have almost same probabilities, given WA). The probabilities in this version are the same as in the original version, and the ensuing developments are the same as well: the rebels took over (E), but, despite A9, decided to instruct the US to relinquish the base, which the US did (C). Surely A9 is no cause of C, intuitively. Indeed, on my analysis, the case is one of aipi, with E a decreaser, and no strict increaser. Hence, on my analysis A9 is not a cause of C. However, A9 increases the probability of C. Further, on Lewis’s analysis, this should be so counterfactually as well: the chance of C given Wi;AO 9 (for a ;A9possible world Wi with the actual history WA) is supposed to be much lower than 420 / Igal Kvart in the actual case, i.e., given WAO 9. Hence, on Lewis’s analysis, A9 is supposed to come out as a cause of C, which is counterintuitive. This failure of Lewis’s counterfactual probability increase condition as a sufficient condition for being a cause is a consequence of its failure to take into account the intermediate course WAO 9, C. (The lack of awareness of the problem posed by decreasers plagues Menzies’s account too, regardless of whether he is successful in his attempt to save Lewis from the problem of the causal irrelevance of c to e. Attempting to overcome the consequence that on Lewis’s account the causal relevance of c to e is not assured, Menzies suggests a condition requiring that, for any series of times between c and e, there be a series of events occurring at those times such that, with c and e added to the series, there be probabilistic dependence between any two consecutive members of the series.96 A9 ends up being a cause on Menzies’s improved version of Lewis as well, since A9 is indeed causally relevant to C: one can follow a trail of events involving actions and states-of-mind of decision-makers in the State Department,97 where each stage increases the probability of the next, counterfactually (a la Lewis’s) or straightforwardly. Thus, Menzies’s modification, designed to secure causal relevance, is no protection against the neglect of decreasers and, ultimately, strict increasers: such a trail need not include such external events as the rebel takeover, which is a strict increaser here.) Salmon’s second approach, that of successive reconditionalization, has the same failing. Probability increase through reconditionalization may nonetheless suffer from lack of causal relevance due to causal breakers, or alternatively may fail to confer the status of being a cause due to the presence of decreasers either between c and e (when c increases the probability of e), or between two consecutive members of the corresponding series (in the absence of strict increasers). In this respect, the analyses of both Lewis and Salmon are plagued by the same problem. The above variation on the dictator example brings out Salmon’s inattention to strict decreasers, which he shares with Lewis: A9 is not a cause of C, but A9 increases the probability of C, satisfying Salmon’s sufficient condition for being a cause. Indeed, Rosen-style counterexamples to a Suppes-style approach highlight the possibility of being a cause despite probability decrease. But Suppes’s approach and subsequent refinements, including Salmon’s and Lewis’s, seem content with the leading idea that probability increase is a sufficient condition for being a cause. As the above example shows, however, this is not the case. The same problem also further plagues both Lewis’s causal series approach and Salmon’s successive reconditionalization condition due to their reliance on cause transitivity. We saw above that cause transitivity is invalid.98 But putting this aside, consider chains satisfying Lewis’s causal series requirement or Salmon’s successive reconditionalization condition. The sufficient condition of counterfactual probability increase must obtain between any two consecutive members (for Lewis), and similarly a straightforward conditional probability increase condition must obtain between any two consecutive members (for Salmon). Yet such series are open to the same vulnerability that the overall sufficient condition itself Cause and Causal Impact / 421 faces. The relation between two consecutive events in such a chain need not render the first a cause of the second, despite counterfactual or straightforward probability increase, since there may be either a causal breaker or a strict decreaser between them for which there is no strict increaser. Thus, Lewis’s and Salmon’s accounts would fail regardless of the failure of cause transitivity: not only due to the obstacle that, despite their conditions, c might not be causally relevant to e, but also because there may be decreasers between c and e or between two consecutive members of a causal series designed to ensure that c is a cause of e, decreasers that would nonetheless forestall c’s being a cause of e. The main problem recognized by adherers of probabilistic analyses of cause via the conception of raising the probability of the effect was that c can be a cause of e even when c lowers its probability. What the above example and the theory of decreasers and increasers brings out is that c can also fail to be a cause of e even when c does indeed raise the probability of the effect: raising the probability is not a sufficient condition for being a cause. The idea that it is a sufficient condition has long been a basic tenet of probabilistic analyses. While the central response to the failure of ‘raising the probability of the effect’as a necessary condition of being a cause has been to resort to series of intermediate events which fulfil the condition of raising the probability pairwise through transitivity (e.g. Salmon, Lewis), no similar move is available to deal with this mirror problem of the insufficiency of ‘raising the probability of the effect’. 16: The Inadequate Combination of Lewis’s Counterfactual Account and His Notion of Chance-Consequent Counterfactuals. Thus far I have treated Lewis’s analysis of cause under the pretense that, aside from the problems I indicated, it functions more or less as intended, at least in straightforward cases of probability increase or probability decrease. But the analysis faces another fundamental problem, one which pulls the rug out from under that pretense. The variant of the dictator example can illustrate this problem facing Lewis’s conception of counterfactual probability increase as providing a sufficient condition for being a cause. Suppose that A9 (the US refused to send arms) occurred, the probabilities as of tA9 are the same as before, but E did not in fact occur, and C did—the US relinquished the base (at the request of the dictator, on account of A9 ). So A9 is a cause of C. A9 surely increases the probability of C—aipi. Suppose the actual chance, P(C/WAO 9), is 0.7.99 On Lewis’s counterfactual analysis, this actual chance must be compared with the hypothetical chance in alternative possible worlds in which ;A9 holds— i.e., roughly, A.100 In such worlds, the US decided to send the arms, and indeed the probability of C given A and WA is very low. However, suppose that an event with a thrust similar to E’s took place in one such possible world W*, but at tA9 —call such an event E9. Given WA, E9 (happening at tA) 101 may be highly unlikely, but it may well be compatible with WA, and such compatibility is all that is needed for 422 / Igal Kvart our purposes. At the very least, suppose some event other than E9, call it E99, happened at tA which made the rebels’ success soon thereafter very likely (as opposed to the chance of the rebels taking over during tA, which was, given WA, very low). For instance, under a suitable fleshing out of the example, if the dictator dies at tA, which can plausibly be seen as compatible with WA, the chance of the rebels taking over may become high indeed. Hence, in such a W* (in which A occurs, namely, the US decided to send arms to the dictator), the chance of C as of tAO (as these chances should be relativized according to Lewis) would be even greater than the actual chance of C given WAO 9 (and thus A9 )—say, 0.9. On Lewis’s analysis, therefore, there are selected possible worlds with an even higher conditional chance of C than the actual world; hence the actual conditional chance is not ‘much higher’—but rather, even lower—than the supremum of the range of conditional chances in the selected possible worlds (i.e., {P(C/WiAO )}). Hence on Lewis’s account, there is no counterfactual probability increase after all, and Lewis’s sufficient condition doesn’t hold. The type of situation illustrated by this example is pervasive: generally, or at least in a wide range of cases in which c is a cause of e, there is no counterfactual probability increase as Lewis claims, since something unlikely (possibly highly unlikely) may happen during tc so that in a ;c-world W9 in which this happens, the conditional chance (given W9;cS , which includes ;c) is higher than the conditional chance (given WcS , which includes c) in the actual world W. Lewis’s sufficient condition for cause is, in fact, often not satisfied when c is a cause of e, and his articulation of the notion of counterfactual probability increase in such cases is ill-suited as a notion of probability increase. Although Lewis considers his conception of counterfactual probability increase to be advantageous in comparison with straightforward probability increase, the problem for his account presented here seems to render this perceived advantage highly questionable. Lewis cannot fall back on cause transitivity (even were it valid), since the problem can arise nonetheless, namely, between the stages within the causal series. We are concerned here with cases of c’s intuitively being a cause of e, where direct counterfactual probability increase fails due to the above consideration. But indirect counterfactual probability increase can also in general be expected to fail, since one can expect that selected possible worlds will often be found in which highly unexpected (very low chance) events during tci (for a cause-candidate ci in the candidate for a causal series) would foil the counterfactual probability increase between the stages. The conclusion seems to be that Lewis’s counterfactual conception of probability increase is not viable. Menzies suggested (in a way unrelated to my critique above) that in the condition not WcS but Wc • c (to use my notation) be considered, as in my treatment (in the actual world W, and mutatis mutandis in other possible worlds).102 But this collapses Lewis’s account of chance-consequent counterfactuals (under his construal of counterfactuals in an indeterministic world) into the probability increase condition (my version): it becomes identical with aipi. The reason is that, given Lewis’s account of such counterfactuals, the only constraint Cause and Causal Impact / 423 placed on the selected possible worlds is that they be ;c-worlds which share Wc and the laws with the actual world, and the chance of the consequent e in every selected possible world is simply (according to Menzies’s modification) the chance of e given Wc • ;c (contrasted with Wc • c, in the actual world). Thus, on this modification, which would often save Lewis’s counterfactual probability increase from not holding when it should (as indicated above), there are no numerous possible worlds to consider separately, since in the selected possible worlds only Wi;c is preserved: it is the only history taken into account. Hence, Lewis’s theory collapses into aipi as a sufficient condition for being a cause. Obviously, given the arguments presented above, aipi is a highly inadequate sufficient condition. And once augmented by cause transitivity, it becomes akin to Salmon’s successive reconditionalization framed in terms of chances (instead of Salmon’s relative frequencies). Hence Lewis’s counterfactual probability increase account either fails as a notion of probability increase, if taken as formulated by Lewis, or, if modified a la Menzies, reduces to Salmon’s second theory adapted in terms of chances. 17: Salmon’s Counterfactual Circularity Salmon’s first strategy, that of reducing causes to counterfactuals, can now be seen to be circular. His two main conditions, MT and CI, as noted, rely heavily on counterfactuals. If my analysis is correct, counterfactuals invoke causes. (More specifically, they require the preservation of intermediate events of which the antecedent is not a cause). Without an independent notion of cause which doesn’t itself rely on counterfactuals, to invoke counterfactuals for the analysis of cause is therefore circular. My own analysis of the notion of cause does not rely on counterfactuals, but is, rather, a purely probabilistic analysis; its constituents play an important role in the analysis of counterfactuals (specifically, the notions of causal irrelevance and purely positive causal relevance). Salmon conceives causes in terms of causal processes and causal interactions, but in requiring counterfactuals for MT and CI he, thereby, if my account of counterfactuals is correct, invokes causes.103 As we mentioned, Salmon neither offers nor endorses a theory of counterfactuals, save to express the hope that an adequate theory will be found. Thus, if my account of counterfactuals is correct, Salmon’s project of securing the notion of cause through causal interactions and causal processes while relying on counterfactuals is circular. He falls short of solving Hume’s puzzle, not just due to extensional inadequacy, and not just in view of the nomic aspects of counterfactuals, but due to the outright circularity of the project. Salmon expresses the hope that the role of counterfactuals in pertinent experiments will bring out their objective character, and will make it plausible that an objective analysis of counterfactuals can be invoked for his analysis of causal processes. Although my analysis fulfils Salmon’s hopes regarding objectivity, the hope that counterfactuals can be invoked for Salmon’s analysis in a non-circular way is, as we argued above, unwarranted.104 424 / Igal Kvart Notes 1. Lewis emphasizes that “[his] analysis is in terms of counterfactual conditionals about probability; not in terms of conditional probabilities.” Cf. David Lewis, Philosophical Papers, vol. II, Oxford University Press, 1986, p. 178. 2. David Lewis, Counterfactuals, Harvard University Press, 1973. 3. See Kit Fine’s review of Lewis, Mind 84 (1975), 451–8. 4. David Lewis, “Counterfactual Dependence and Time’s Arrow”, Nous 13 (1979), 418– 46; reprinted in Lewis’s Philosophical Papers, vol. II, op. cit, pp. 32–52. 5. In his Philosophical Papers (1986) Lewis denies that his “Counterfactual Dependence and Time’s Arrow” (henceforth: CDTA) presents a new theory, as opposed to that put forward in his 1973 book); see Philosophical Papers, vol II, pp. 52–3. Yet he is aware that not a few of his readers think otherwise. In his book, Lewis uses the term ‘familiar notion of comparative overall similarity’: it is by means of this notion that he analyzes counterfactuals (Lewis, Counterfactuals, op. cit, p. 92). I have used the term ‘intuitive overall similarity’. In Postscript A (Philosophical Papers, vol. II, p. 53) Lewis uses, instead, the term ‘explicit (or offhand) snap judgment’, and agrees that the similarity relation he put forward in CDTA is not this relation. The revision, as I see it, consists in abandoning intuitive overall similarity and replacing it with a notion governed by respects and priorities that do not yield intuitive overall similarity (as attested by the outcome in the case of Fine’s counterexample), and which must be specified by constraints on the respects and priorities designed to fit the intuitive truthvalues of particular counterfactuals (cf. CDTA, Philosophical Papers, vol. II, p. 42). Whereas Lewis’s original theory is to a considerable extent testable, the theory of CDTA provides rules for priorities designed specifically to handle Fine’s counterexample, but does not purport to be complete, and does not provide information regarding other respects and priorities-constraints of this non-intuitive similarity relation; see CDTA, ibid, p. 48. (Note, for instance, that Lewis, later on, adds quasi-miracles as a weighty dissimilarity respect; cf. ibid, p. 65.) For a detailed discussion and critique of Lewis’s CDTA theory, see my A Theory of Counterfactuals, Hackett, 1986, ch. 8. 6. In a deterministic world some miracle or other must be invoked to accommodate the antecedent. 7. David Lewis, Philosophical Papers, vol II, p. 181. In his “Counterfactual Dependence and Time’s Arrow” he is more hesitant (op. cit., p. 48). 8. David Lewis, Philosophical Papers, vol II, Postscripts to “Causation” (pp. 172–214), in particular Postscript B (“Chancy Causation”, p. 175), and especially pp. 180–2. 9. Ibid, p. 59. 10. For my approach regarding backtracking, compare my “Counterfactual Ambiguities, True Premises and Knowledge”, Synthese 100 (1994), 133–164. 11. Cf. p. 201; p. 179; p. 242. This leaves a lacuna, since some cases, in which c is a cause of e, are such that ;c is incompatible with Wc, the world history up to tc (the time to which c pertains). For instance, take as e ‘the bullet hit the wall at spot S at time t1’ (S and t1 are highly specific spot and temporal instant), and as c ‘the bullet is in region R at time t9, where R is broad enough so that ‘the bullet is not in R at t9 is incompatible with Wc (which includes the trajectory of the bullet until tc). Still, c is a cause of e. Lewis also counts quasi-miracles as weighty dissimilarities from actuality (p. 63), but this seems to have little bearing on our discussion. 12. I assume that the counterfactual is assessed in the actual world. Cause and Causal Impact / 425 13. That is, the only constraint is perfect match until this interval tc ; see Lewis, 1986, pp. 180–1. (This account makes the counterfactual construction objective indeed.) Lewis has been consistent in not recognizing the importance of intermediate events. In his “Counterfactual Dependence and Time’s Arrow” (1979) he discusses what he calls Analysis 1 of counterfactuals, which he attributes to various authors; it requires the preservation of a prior history and of the laws of nature (after tA; cf. his Philosophical Papers, vol. II, p. 39). He rejects this analysis for reasons unrelated to our concerns here, but does not find it flawed because it is independent of the intermediate history (that between the antecedent and the consequent). 14. This account is thus a possible-world counterpart of an inferential-model account on which a counterfactual A.B is true iff the consequent B follows (by the laws) from WA plus A. (WA is the world history up to tA.) 15. To facilitate discussion of Lewis’s account, I follow his notation, with c as a causecandidate and e as an effect-candidate. Beyond the discussion of Lewis, I revert to my usual notation of A and C respectively. 16. I employed it for the analysis of pertinent causal notions in my Ph.D. dissertation (1975, University of Pittsburgh). It was elaborated on in my 1986 A Theory of Counterfactuals (henceforth ATC), Hackett (currently published by Ridgeview). 17. Philosophical Papers, vol II, op. cit., p. 177. 18. The notation employed here is mine. 19. See Lewis 1986, p. 176–7. 20. Cf. Lewis, ibid, p. 167; p. 242. In particular, Lewis needs cause transitivity to handle cases of causes with probability decrease (p. 179) and cases of preemption (p. 200). 21. Princeton University Press, 1984. 22. Wesley Salmon, ibid, p. 148. I limit my discussion here to Salmon’s book. Salmon recently changed his position by abandoning the counterfactual component of his process theory and adopting a conserved quantities account; see his “Causality Without Counterfactuals”, Philosophy of Science 61 (1994) 297–312. Salmon was responding to a constructive (as well as critical) rejoinder by Dowe (and earlier critiques). Dowe’s account replaces Salmon’s definitions of MT and CI by concise definitions which appeal to the notion of a conserved quantity and do not invoke counterfactuals. See Phil Dowe, “Wesley Salmon’s Process Theory of Causality and the Conserved Quantity Theory”, Philosophy of Science 59 (1992), 195–216. On Salmon’s new analysis (1994), a causal process transmits an invariant quantity, in accordance with Definition 3 (p. 308): A process transmits an invariant (or conserved) quantity from A to B (A Þ B) if it possesses this quantity at A and at B and at every stage of the process between A and B without any interactions in the half-open interval (A, B] that involve an exchange of that particular invariant (or conserved) quantity. But Salmon has by now become anxious to avoid counterfactuals due to criticisms other than the central criticism I put forward here (see below, sections 13 and 18), the latter being that invoking counterfactuals yields circularity due to reliance on the notion of cause. To avoid counterfactuals, Salmon is forced to interpret the locution “without any interactions”, which would normally call for a counterfactual interpretation (as in Salmon’s MT), indicatively. Salmon is aware that, realistically, processes undergo interactions at least much of the time (cf. Philip Kitcher, “Explan- 426 / Igal Kvart 23. 24. 25. 26. 27. 28. 29. 30. 31. atory Unification and Causal Structure”, in Scientific Explanation, P. Kitcher and W. Salmon (eds.), Minnesota Studies in the Philosophy of Science, vol. XIII, 1989, p. 463). But the price exacted by this move is high: the notion of a process becomes degenerate. What Salmon previously (e.g., before this emendation) called processes become tiny, disconnected, non-contiguous process-segments between interactions, in the small, and in the case of macroscopic objects in the real world (i.e., not in vacuo), the notion of a process becomes almost inapplicable and thus empty. With regard to the case of a gas molecule, for example, the molecule-process turns out to be causal between interactions, on this definition. Consequently, Salmon presumably would extend his definitions to the effect that a process all the segments of which (other than those undergoing interactions) are causal, would also be considered causal. But what about the speeding bullet, with its interactions with air molecules? Here Salmon says, “we would simply ignore such interactions because the energymomentum exchanges are too small to matter.” (p. 309). But this resort to idealization will not do: while Salmon counsels that we ignore interactions with air molecules and, as a good enough approximation, regard the speeding bullet as causal, what counts for the analysis of a causal process is not the idealized perspective, but rather the reality that in fact it is not, which is intuitively the wrong outcome, even if under idealized conditions it would be. Even worse: consider, instead of a speeding bullet, a stone sinking in the ocean. Here, the friction with the water molecules is not negligible (if desired, examples with denser liquids can be exploited), so that no resort to idealization is available, yet the process, intuitively, is indeed causal. Hence Salmon’s ploy of invoking a non-counterfactual condition for transmission, and thus for the characterization of causal processes, does not succeed. In the absence of interactions with other process, P would continue to exhibit Q. See the last note. This qualifying clause invites a counterfactual construal. Salmon, Scientific Explanation and the Causal Structure of the World, ibid, p. 171. Ibid, pp. 149–50. For my analysis of the Bizet-Verdi example, see ATC, p. 237. My theory of counterfactuals would accommodate Salmon’s concern, since counterfactuals associated with scientific experiments of the sort Salmon discusses (p. 150) do indeed come out objective on my theory. I have argued that non-backtracking counterfactuals (n-d counterfactuals) as well as back-tracking counterfactuals (l-p counterfactuals) are largely objective, except for isolated types with context-dependence that is mostly not specific to the counterfactual construction. Cf. A Theory of Counterfactuals, esp. ch. 9, IV, 4 (pp. 254–256). For my analysis of another paradigmatic case of so-called context-dependence (on Lewis’s view—see CDTA, p. 34, following Quine), namely, that of Caesar and the catapults, see A Theory of Counterfactuals, ibid, p. 247. Cf. also J. Fetzer, “Critical Notice: Wesley Salmon’s Scientific Explanation and the Causal Structure of the World”, Philosophy of Science, 54: 597–610, 1987. Salmon, pp. 178, 182. He says that when we think of the cause-effect relation, we typically, but not invariably, think of such cases. Cf. also Philip Kitcher’s discussion of Salmon’s paradigm in his “Explanatory Unification and Causal Structure”, op. cit., p. 461. Salmon also characterizes the notion of a cause simultaneous with its effect in terms of a causal interaction between two processes where one leaves a mark on another; I will not get into this notion here. So long as the temporal order is appropriate. As another instance of this shortcoming, consider a ball on its way after being thrown. While it is in the air, projector #1 was lit and cast a beam of light on the ball at t1; Cause and Causal Impact / 427 32. 33. 34. 35. 36. 37. consequently, the ball heated up a bit. This is a causal interaction between P (the ball) and P9 (the beam of light, another causal process). After a short while, at t2, at another location, another projector #2 was lit and cast a second beam of light on the ball. (The second projector was independent of the first.) The first interaction is no cause of the second; in fact, the first interaction between the ball and the light is causally irrelevant to the second. But this pair of causal interactions fall within Salmon’s paradigm of cause and effect. (For a related point, see Kitcher, who goes on to argue that Salmon needs a much heavier reliance on counterfactuals than manifested in MT and CI; Kitcher, op. cit., sec. 6.3.) Dowe’s account of cause (in his 1992 paper), assessed by the examples he provides (p. 211), seems open to the same charge even though he, like Salmon, avoids spelling out there an explicit analysis of cause. Providing an account of causal interactions will, if adequate, indeed be a major contribution to the characterization of cause-effect pairs straddling such interactions. But this subset of cause-effect pairs covers only one type of such pairs, albeit a central type. As a central part of his paradigm for the cause-effect relation, Salmon suggests the requirement that there be a connecting process (between two causal interactions). But this isn’t necessary: P9 ’s interacting at c with P can be a cause of P99 ’s interacting with P99 at e after P99 interacted with P at d, even without one single causal interaction connecting c and e. Salmon accepts cause transitivity for his later theory of successive reconditionalization (see below); so presumably he would accept it here too. But not any three such interactions preserve a cause-effect relation—see the failure of cause transitivity below. Salmon may be able to develop his notion of a causal process as something that carries causal propensity, so that hitting the ball first produces a causal propensity (energy, direction of momentum) that propagates to the second interaction (the ball’s breaking the window). But he does not seem to do so, and only mentions it briefly at the end of chapter 7, in a different context—that of probabilistic causation. Dowe’s constructive account, while avoiding Salmon’s reliance on counterfactuals and numerous difficulties associated with conditions MT and CI, appears to suffer from the same shortcomings as Salmon’s account insofar as the characterization of cause is concerned, in that the notions of causal process and causal interaction, even if adequate and non-circular, do not go far enough. Dowe offers no additional analysis of the notion of cause, beyond characterizing certain types of cases (and an example— (3); op. cit., pp. 213–4). But the lack of a general analysis of the basic distinction between cause-effect pairs and other pairs, pointed out above with regard to Salmon, is not resolved by Dowe’s definitions and discussion, since this distinction cuts across the types of cases he mentions. (Again, note the possible exception of the important case of cause-effect pairs straddling a causal interaction, which, understandably, seems to be the focus of their interest.) This line of thought typically employs relative frequencies. c is: the ball was hit in the direction of the branch. Given ;c, with high probability the ball would be hit in the direction of the hole. Salmon doesn’t endorse this account in the end, because it can’t handle quantum mechanical events in discrete rather than dense or continuous time, where there need not be intermediate events. This doesn’t seem to me a good reason for rejecting the account: a theory that adequately handles causes in the macro world is an important feat in itself. Quantum Mechanics presents huge problems regarding causes which involve very strong violations of our intuitions, and our ordinary, intuitive, macro notion of cause 428 / Igal Kvart 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. may indeed prove to be inapplicable to the quantum-mechanical phenomenon. I do not take my own account of cause to be applicable to quantum-mechanical cases, and for the same reason I do not consider this fact to undermine my account. E.g., along the lines of ordered triples of an object, a predicate (or property) and a time, for a simple case; or alternatively, taken under descriptions. See also note 98 below. Cf. also Eells’s discussion of non-transitivity of ‘because’. But note that in this paper the subject is causes, not ‘because’. My discussion therefore focuses on the indeterministic case. Cf. my ATC, ch. 2, esp. V and VI, and my “Counterfactuals”, Erkenntnis 36 (1992), 1–41. I also assume that the indicative form of the antecedent and consequent are factual. tBO is the upper limit of the interval tB. A (B, C etc.) will play the double role of being both statements and the events they specify. To avoid cumbersome formulations, I allow myself to casually invoke statements when, strictly speaking, it is the events they specify that are being discussed. (;A is causally irrelevant to E just in case the actual A is.) WA, BO is the world history from the beginning of tA till the end of tB. A non-actual A will be considered as having purely positive causal impact on E just in case the actual ;A has purely negative causal impact. The inferential schema for counterfactuals, and with it truth values for counterfactuals, can be replaced with high objective conditional probabilities, where the probability of the consequent is conditional on the antecedent and the factual implicit premises, that is, the prior history and the intermediate events in (tA, tBO ) to which A is causally irrelevant or purely positively causally relevant. The importance of these intermediate events is undiminished in this change of the semantic value of counterfactuals from truth value to conditional objective chance. I do not, however, dwell here on this conception and the reasons for favoring its adoption. Similarly, F has purely negative causal impact on E just in case F has some negative causal impact on E but no positive causal impact at all. Examples of this sort were examined in my ATC (examples 3 and 8, ch. 2). A is causally irrelevant to C iff C is causally independent of A. I assume that the stock was sold in a private transaction (that remained confidential at least until t3) to an individual who, in turn, was not interested in selling and in fact did not sell (until t3). The following day. Ibid, p. 39. Cf. ATC, ch. 2, sec. VIII. The approach I pursued is not circular. Of course, on my view, reducing causes to counterfactuals would not qualify as a reductive analysis, since counterfactuals in turn hinge on causes (section 13). Cf. note 82 below. In this paper I do not distinguish between causal relevance and causal impact— between A’s having causal impact on C and A’s being causally relevant to C. Surely, to have causal impact is to be causally relevant. A more detailed discussion cannot be undertaken here. See note 44. The analysis of cause here is thus obviously geared toward an indeterministic world. I used this notion in my Ph.D. dissertation (1975, University of Pittsburgh). For more on this notion of probability, see my “Counterfactuals and Causal Relevance”, Pacific Philosophical Quarterly 72 (1991), pp. 314–337, section 4, and my ATC, ch. 4, sec- Cause and Causal Impact / 429 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. tions I and VI. For related conceptions, cf. K. Popper, The Logic of Scientific Discovery, Science Edition, Inc., 1961; I. Hacking, The Logic of Statistical Inference, Cambridge University Press, 1965; H. Mellor, The Matter of Chance, Cambridge University Press, 1971; B. Skyrms, Causal Necessity, Yale University Press, 1980; and D. Lewis, “A Subjective Guide to Objective Chance”, in Ifs, W. Harper, R. Stalnaker and G. Pearce (eds.), Reidel, 1981. We need to require causal relevance for the relation of some positive causal impact since otherwise, even if there is a strict increaser for A and C (see below), there may still be a causal cutter, i.e., an event that cuts off the would-be causal connection between A and C and renders A causally irrelevant to C. Here I rely on the probabilistic analysis I offered for the notion of causal irrelevance; see my “ Transitivity and Preemption of Causal Impact”, Philosophical Studies, 64 (1991), pp. 125–160; and my “Causal Independence”, Philosophy of Science 61 (1994), pp 96–114.) WA, C is the world history pertaining to the time interval between tA and tC, i.e., (tA, tC). (tC is the beginning point of the interval tC.) Cf. my “Counterfactuals and Causal Relevance”, Pacific Philosophical Quarterly 72 (1991), 314–337, for the analogous distinction between expected and de facto causal relevance and the importance of WA, C. (tA, tC) is the time interval which starts at the beginning of the interval tA and ends at the beginning of the interval tC. tA is the beginning of tA. I.e., in view of all the intermediate facts (and probabilistic relations). Arguably, in this case A does not have any negative causal impact on C. The probabilities as of t1 (and their assessments by the US) are as before. A9 is very close to ;A though not quite the same (but with almost, we assume, the same probability as of tA). Assume that the portfolio handling arrangement was for transactions to be executed on specific instructions to that effect. tA5t;A; hence WA5W;A. The empty event in (tA, tC) can be considered an increaser. Note the function of the preliminary condition requiring causal relevance: a strict reverser E can have a blocker (i.e., an intermediate event which, when added, together with E, to the condition on both sides, yields equality; cf. “Causal Independence”), and thus, in the absence of an assumption requiring causal relevance, is compatible with causal irrelevance. A strict increaser is, then, a sufficient condition for spci only if there is causal relevance. This thesis applies to the notion of strict increaser taken to include null strict increasers as well as strict increasers in aipi cases (as will be explained in section 11), and is extended below (see especially note 75). Note that a corresponding modification applies to the definition of the notion of a strict decreaser, which consequently yields an analogous extension of its definition. The extension of the definition of strict increaser here (and of strict decreaser as per the previous note) constitutes a change in the terminology I used in earlier writings, where I suggested the notion of a clincher (increaser-clincher and decreaser-clincher, respectively). Though the current usage is simpler for my purposes here, the difference is merely terminological. Cf. my “Overall Positive Causal Impact”, Canadian Journal of Philosophy 24 (1994), pp. 205–228, especially p. 220. Note too that given this terminological extension, strict increasers and strict decreasers are compatible, and can co-exist. The presence of a strict increaser establishes 430 / Igal Kvart 75. 76. 77. 78. 79. 80. 81. 82. spci, of a strict decreaser, snci. Obviously, spci and snci can and often do co-exist, and so the co-presence of a strict increaser and a strict decreaser is thus common. See my “Overall Positive Causal Impact”. One must add, though, that the notion of spci, and consequently of being a cause, are closed under informational (e.g., conjunctive) expansions. Thus, A • D may be a cause of C when there is a strict increaser for A and C even when there is no strict increaser for A • D. See my “Causes and strict increasers” (forthcoming). But see the qualification in the previous note. See, for instance, Lewis, op. cit., p. 179. See note 41, and the parenthetical comment after Analysis (I), section 6. See note 46. A is a cause of C in case A is causally relevant to C and there is a strict increaser. A is not a cause of C when A is causally irrelevant to C or else, causally relevant and there is no strict increaser. Indeed, there can be a strict increaser even when there is causal irrelevance (e.g., due to a causal breaker). E.g., by characterizing such cases (for two actual or potential causes) by the counterfactual ;c1 • ;c2.;e (even though ;ci.;e is false, i51,2). See section 17 below for a related point regarding Salmon’s project. Lewis’s program of reducing causes to counterfactuals, interestingly enough, largely escapes the brunt of the circularity charge, since the counterfactuals he employs in his analysis of cause are very special ones: chance-consequent counterfactuals, i.e., counterfactuals with consequents specifying chances of certain events which do not depend on intermediate events, and thus the issue of the causes of such intermediate events does not arise (nor do those of causal irrelevance and purely positive causal relevance to such events). Were we to assume that Lewis’s analysis of cause is extensionally adequate, we could conclude that it largely bypasses the need to resort to the notion of cause in the analysis of counterfactuals (on my account) by limiting the use of counterfactuals exclusively to chanceconsequent counterfactuals, and thus largely avoiding the circularity issue. In making this observation, I assume the adequacy of my analysis of n-d counterfactuals. Of course, invoking Lewis’s own account of counterfactuals in his analysis of cause is free of circularity, since surely his own account of counterfactuals does not invoke causes. In his 1986 account of counterfactuals in an indeterministic world, Lewis held out against preserving any intermediate event after tA, retaining only WA. In this respect his approach was consistent with his earlier views on the subject, and on this issue there is a fundamental difference between his approach and my own. Circularity arises primarily when the counterfactuals employed are fullfledged counterfactuals, i.e., unlike chance-consequent counterfactuals, which are independent of the actual history during (tc,e S ). To recapitulate: The impression that Lewis’s counterfactual analysis of cause is circular would be based on my contention to the effect that counterfactuals invoke causes (in the implicit premises), and hence an analysis of cause in terms of fullfledged counterfactuals would face the charge of circularity. Note, though, that Lewis’s analysis of cause in an indeterministic world need not be conceived as a counterfactual analysis, since it can be expressed directly in terms of possible worlds, in accordance with the constraints he employs for counterfactuals. Thus, his analysis of cause in the indeterministic case can be framed solely in terms of chances, by comparing the appropriate conditional chance of e in the actual world with this conditional chance in selected worlds. So presented, it need not be viewed, strictly speaking, as an analysis Cause and Causal Impact / 431 83. 84. 85. 86. 87. 88. 89. in terms of counterfactuals, and thus the possible-worlds segment of his theory can be claimed to be immune to the charge of circularity. Conceiving his account of cause as a counterfactual analysis is quite natural, since Lewis aspires to unity in his overall analysis of cause (covering both the deterministic and the indeterministic cases), and indeed, by conceiving his analysis of cause in the indeterministic as well as the deterministic case to be counterfactual, the analysis being unified appears plausible. But his resort to counterfactuals in the indeterministic case is, to a certain extent, somewhat degenerate due to the lack of dependence on Wc,e S , brought about by invoking chance-consequent counterfactuals, rather than regular counterfactuals. Moreover, in view of his aspiration for a unified analysis, it is natural for Lewis to conceive his account in the indeterministic case as one of counterfactual probability increase, since on his account, c is a cause of e just in case the conditional chance had c been actual would have been much greater than had c not been actual. In the actual world he can present the pertinent conditional probability through the counterfactual: c . P(e/WcS )5x. In the hypothetical case, the chance in each world has to be considered separately, and there is no one value of the conditional chance in all the selected worlds. The counterfactual element is manifest when formulated as follows: Had ;c, then the conditional probability would have been much lower than x; i.e.: ;c . P(e/W;cS ),, x. This presentation in a counterfactual form allows Lewis to claim to have produced a unified analysis. Lewis can therefore have it both ways. But, as I will argue (section 16), this analysis, whether in the counterfactual form, or presented, as described above, without counterfactuals, is untenable, since the condition for being a cause is generally not fulfilled even when the cause-candidate is indeed a cause. Were Lewis’s conception modified, as Menzies suggests, to have the chance temporally relativized to tc (rather than to tcS , as on Lewis’s account) in the pertinent chance-consequent counterfactuals, they would collapse into conditional chances, and the counterfactual form would become largely degenerate. The reason is that the counterfactual c . P(e/WcS )5x in Lewis’s original version becomes c . P(e/Wc • c)5x, since c must be part of the condition here; similarly, ;c will have to be inserted in the condition in the counterfactual with the antecedent ;c, which will become: ;c . P(e/W;c • ;c),, x. But W;c, for the selected worlds, is just Wc. Further, c . P(e/ Wc • c)5x is equivalent to P(e/Wc • c)5x, and ;c . P(e/Wc • ;c),,x is equivalent to P(e/Wc • ;c),, x. Hence the entire condition boils down to P(e/Wc • ;c) ,, P(e/ Wc • c), which is very close to aipi (the difference being ‘,,’ rather than ‘,’). And aipi, as we noted, is a highly inadequate account of cause. As throughout, I assume an indeterministic world. On Lewis’s account and on my own. Stalnaker’s conclusion is different. Cf. R. Stalnaker, “A Defense of Conditional Excluded Middle”, Ifs, W. Harper, R. Stalnaker, G. Pearce (eds.), Dordrecht: Reidel, 1980. Cf. Lewis’s related observation, Philosophical Papers, vol. II, p. 176. Assume that without #1, the other two might have increased their own efforts. Counterparts of cases which, in a deterministic world, are over-determined, need not be so in an indeterministic world. As noted, only the indeterministic case is considered here. Multiplicity of contributing causes is a standard case rather than an esoteric exception. I do not deal here with the notion of being the cause, a notion that dominates many legal discussions of causation. Cf. Hart and Honore, Causation in the Law, London: Oxford University Press. 432 / Igal Kvart 90. Not even overall positive causal impact: cf. my analysis of this notion in “Overall Positive Causal Impact”, op. cit. 91. Cases with potential causes are more amenable to the efficacy of cause transitivity in protecting ;A.;C as a necessary condition of being a cause. 92. I take the purely positive causal impact of ;A on C to be tantamount to the purely negative causal impact of A on C. 93. Through purely positive causal impact. 94. An observation to a similar effect was made by Menzies; see his “ Probabilistic Causation and Causal Processes: A Critique of Lewis”, Philosophy of Science 56 (1989), 642–663). In my Ph.D dissertation (University of Pittsburgh, 1975) and in ATC (1986), I offered an account of causal irrelevance that was also independent of intermediate events, and thus vulnerable to causal breakers (events that cut would-be causal chains), a danger I was well aware of when I stated: “ The analysis of causal irrelevance in chapter 4 is a function of WtA only, and as such is a mere approximation of the notion of causal irrelevance requisite for the analysis of counterfactuals. An attempt to provide an analysis sensitive to (tA, tC) as well will not be made in this book and will be left for another occasion.” (ATC, Errata, p. 126, fn. 22.) The reasoning underlying this comment was expanded on in my “Counterfactuals and Causal Relevance”, Pacific Philosophical Quarterly 72, 4 (1991), 314–337 (featuring in particular the possibility of causal breakers). 95. E.g., aipi or aipd, on my approach. 96. See the reference in note 94. 97. It may include an intermediate event such as: the state department received a message from the authorities in that country requesting that the US relinquish the base. 98. A caveat is called for: In pointing out the invalidity of cause transitivity in section 5 above, we relied on a narrow individuation of events. Lewis’s conception of event individuation is different, and this problem with cause transitivity is an issue for him only if it applies to his conception of event individuation. 99. It is not higher since, say, other interests played a part in the dictator’s relationship with the US. 100. Let us suppose that the American decisions on whether or not to send arms to the dictator are made publicly, so that for the US not to decide against sending arms to the dictator implies, with very high probability, that the US decided publicly to send arms to the dictator. The difference in content between ;A9 and A is relatively small, for our purposes, as we constructed the example, and we will assume that it does not yield a significant difference for the pertinent chances involved. 101. tA and tA9 are pretty much the same. 102. On this construal of chances a la Lewis’s as modified by Menzies, coupled with Lewis’s construal of counterfactuals in an indeterministic world, the counterfactual ‘if ;c, the chance of e given ;c would be at most y’ becomes tantamount to P(e/ Wc • ;c)5z#y, since all selected possible worlds are the same in the relevant respect of sharing the actual history up to c, Wc, and thus all yield exactly the same chance. 103. Or constituent concepts which together yield the notion of a cause. 104. I wish to thank Martin Bunzl, Alan Hajek, Avishi Margalit, Hugh Mellor, and Richard Otte for their comments.