Abstract
When is a cause of a cause of an effect also a cause of that effect? The right answer is either “Sometimes” or “Always”. In favour of “Always”, transitivity is considered by some to be necessary for distinguishing causes from redundant non-causal events. Moreover transitivity may be motivated by an interest in an unselective notion of causation, untroubled by principles of invidious discrimination. And causal relations appear to “add up” like transitive relations, so that the obtaining of the overarching relation is not independent of the obtaining of the intermediaries. On the other hand, in favour of “Sometimes”, often we seem not to treat events that are very spatiotemporally remote from an effect as its causes, even when connected to the effect in question by a chain of counterfactual or chance-raising dependence. Moreover cases of double prevention provide counterexamples to causal transitivity even over short chains. According to the argument of this paper, causation is non-transitive. Transitizing causation provides no viable account of causal redundancy. An unselective approach to causation may motivate resisting the “distance” counterexamples to transitivity, but it does not help with double prevention, and even makes it more intractable. The strongest point in favour of transitivity is the adding up of causal relations, and this is the point that extant non-transitizing analyses have not adequately addressed. I propose a necessary condition on causation that explains the adding up phenomenon. In doing so it also provides a unifying explanation of distance and double prevention counterexamples to transitivity.
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Notes
I assume that causation is a relation on events, despite being attracted to the view that it is the truth maker for a connective of some sort (Mellor 1995, 2004). For the purposes of discussing transitivity, the difference does not matter, so long as whatever you think causation is supports a corollary of transitivity in the first place. Mellor’s connective clearly does.
Lewis’s original discussion is framed in terms of counterfactual dependence, which imposes the further requirement that \(O(c)\,\square\!\!\!\rightarrow O(e)\). But this follows on Lewis’s semantics for counterfactuals, specifically the Centering Assumption, from the requirement that c and e actually occur (Lewis 1973c, pp. 26–31). The Centering Assumption is not uncontroversial (cf. Nozick 1981; Menzies 2004). Although I am not committed to Lewis’s semantics, I will not challenge the Centering Assumption. Hence it is safe to ignore this point for present purposes, and to assume that—for any pair of events x and y which satisfy the existence and distinctness requirements on causal relata—y counterfactually depends on x iff \(\neg O(x)\,\square\!\!\!\rightarrow\neg O(y)\). Note that Lewis also ignores the complications of his original treatment in later work (Lewis 2004, p. 78; cf. Lewis 1973a, pp. 164–167).
Coady argues that for a more precise individuation of events. If Able had not thrown, then that exact bottle smash would not have occurred; a slightly different one would have occurred in its place. However, this strategy is open to a class of objections more commonly made against Lewis’s influence account—namely, that sometimes exactly the same effect may be brought about in more than one way. Hall, crediting Yablo, imagines that Able has got his hands on a Smart Rock, programmed to hit with exactly the same velocity at exactly the same moment, regardless of small differences in throw (Hall 2004b, p. 237). And I have given the less far-fetched example of an alarm clock, which goes off at 7am regardless of whether I press the button to set it, or my wife does (Broadbent 2007, p. 177).
His second solution, in terms of quasi-dependence, arguably does not rely on the assumption that causation is transitive, as discussed below. But his third solution in terms of influence does (Lewis 2004).
I eschew Lewis’s terminology of causal dependence because it seems to me obviously prejudicial. Who could bring themselves to deny that two events connected by a chain of causal dependence are related as cause to effect? Arguably, using “dependence” to express the truth of a counterfactual is also prejudicial, since “depends” inherits heavy causal overtones from its everyday meaning; but let us put up with that for the moment.
Which is enough to negate the claim that it would not have (Lewis 1973b, p. 2).
Elsewhere I have argued that this is an untenable consequence in legal contexts (Broadbent 2009).
Lewis argues that if we are happy to accept that causes can be redundant, we have no reason to aver in cases of double prevention. But this argument ignores our reason for accepting redundant causation. We have a strong intuition that at least some redundant causes are causes (e.g. Able’s throw in the previous example). If we have a strong intuition not to accept double prevention cases as cases of causation, then we do have a reason to aver in these cases: just as good a reason as we have for calling redundant causes. And most people do seem to be intuitively resistant to accepting that the first and last event in a double prevention case are cause and effect.
I am grateful to an anonymous referee for this suggestion.
So that, where c 1 causes e and c 2 is a preempted back-up, there continues to be some sense in which c 1 raises the chance of e even where c 2 make e just as probable in c 1’s absence. For one proposal see Dowe (2004).
One obvious criticism is that on both accounts the truth of a causal claim is relative to a model, and principles of model selection are very difficult to articulate, as these authors all acknowledge.
This sort of difference-making may not be sufficient for explanatory relevance—there remain many irrelevant differences between fact and foil—but their number is considerably reduced. Moreover, as Lipton points out, this approach helps explain why explanation appears to have both objective and subjective aspects. On this analysis, it is up to us what contrast to explain, but then it is an objective matter whether a given event is a difference between the fact and foil we have selected (Lipton 1990).
If the Reverse Counterfactual is a necessary condition on causation then counterfactual dependence cannot be a sufficient condition on causation, without yielding the counterintuitive result that effects cause their causes whenever redundancy is absent and Lewis’s counterfactual is true.
Hausman himself does not accept that backtracking counterfactuals have determinate truth-values, however.
Indeed at one point Hitchcock almost stumbles into asserting the Reverse Counterfactual: “Intuitively, the falling boulder does not save Hiker’s life because without it, Hiker’s life would not have been endangered in the first place. This is just what is indicated by the absence of an active route from [the boulder falling] to [Hiker surviving]: there is no scenario in which the boulder does not fall and Hiker does not survive” (Hitchcock 2001, p. 295). This is not far from saying that, if Hiker had not survived, the boulder would still have fallen: but Hiker would have omitted to duck.
Unless another valid substitute can be found. On inspection it turns out that the only other hopeful substitutes yield the same consequences that this one does. For example, consider \(A\,\square\!\!\!\rightarrow B\), \((A\&B)\,\square\!\!\!\rightarrow C\), hence \(A\,\square\!\!\!\rightarrow C\). In Boulder, it seems reasonable to say: if Hitchcock had not survived, he would not have ducked. But it does not seem reasonable to say: if Hiker had neither survived nor ducked, the boulder would not have fallen. For the obvious scenario where he dies, and also fails to duck, is one where the boulder hits him. To suppose that the boulder did not fall but that something else entered the picture to kill him would be entirely gratuitous.
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Acknowledgments
I am grateful to Arif Ahmed, Alexander Bird, Peter Lipton and the anonymous referees at this journal for helpful comments, and to the PHG Foundation and the Brocher Foundation for supporting this research.
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Broadbent, A. Causes of causes. Philos Stud 158, 457–476 (2012). https://doi.org/10.1007/s11098-010-9683-0
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DOI: https://doi.org/10.1007/s11098-010-9683-0