1 For their comments upon earlier versions of this paper, I would like to thank Nuel Belnap, John Earman, Richard Gale, Mitch Green, and Wes Salmon; audience members at the University of Pittsburgh, the University of Texas at Austin, and the twenty-fourth annual meeting of the Society for Exact Philosophy; and two anonymous referees for the Canadian Journal of Philosophy.

2 For further discussion of these problems from within the context of a probabilistic theory of causation, see my ‘A Generalized Probabilistic Theory of Causal Relevance,’ Synthese 97 (1993) 335-64; ‘The Mishap at Reichenbach Fall: Singular vs. General Causation,’ Philosophical Studies 78 (1995) 257-91; and ‘The Role of Contrast in Causal and Explanatory Claims,’ Synthese (forthcoming). The current paper complements these earlier papers in at least three ways: it presents the problems divested of the technical garb that makes them inaccessible to some readers; it emphasizes the common thread that is easily lost in the more technical treatments; and it argues that these problems are not unique to the probabilistic theory of causation, but arise within any theory that characterizes causation as a binary relation.

3 ‘Causal Laws and Effective Strategies,’ Noûs 13 (1979) 419-37

4 Probabilistic Causality (Cambridge: Cambridge University Press 1991)

5 There are debates as to whether this set can be determined by purely probabilistic criteria, thus saving the theory from circularity. Cartwright and Eells argue that this is not possible. I have argued ('A Generalized Probabilistic Theory of Causal Relevance’) that although the theory requires some sort of causal primitive, it need not employ the very concepts of positive and negative causation that it aims to explicate.

6 The Chances of Explanation: Causal Explanations in the Social, Medical, and Physical Sciences (Princeton: Princeton University Press 1989), 41-2

7 ‘A Generalized Probabilistic Theory of Causal Relevance’

8 Recall that we are assuming background conditions to be held fixed.

9 ‘Statistics and Causal Inference,’ Journal of the American Statistical Association 81 (1986) 945-60

10 ‘Estimating Causal Effects of Treatments in Randomized and Nonrandomized Studies,’ Journal of Educational Psychology 66 (1974) 688-701. The view is also anticipated by Good, I.J. (‘A Causal Calculus I,’ British Journal for the Philosophy of Science 11 [1961] 309)Google Scholar, who has suggested that claims about the causal relevance of B for A are relative to probability distributions over the alternatives to B. Other theories of causation have taken the causal relation to involve more than two argument places; what is essential to the solution sketched here, however, is not merely the number of argument places, but the relativity of causation to an alternative causal factor.

11 This presents a slight technical problem. Since P(X = x) will be zero for all but countably many values of x, it seems as though the values of the function f will be undefined almost everywhere. There are, however, generalizations of the standard definition of conditional probability that will allow us to have well-defined functions such as f. I prefer Kolmogorov's treatment of conditional probabilities as Radon-Nikodym derivatives. See the appendix to ‘A Generalized Probabilistic Theory of Causal Relevance’ for an overview.

12 Although as I have noted elsewhere (‘A Generalized Probabilistic Theory of Causal Relevance’), only suitable ternary causal claims will succeed in describing such functions. The ternary analysis of causation must take a back seat to the essential communicative role of causal claims, but we will ignore these difficulties here.

13 Other versions of this argument can be found in Suppes, P. A Probabilistic Theory of Causality (Amsterdam: North-Holland 1970)Google Scholar; Eells, E. and Sober, E. ‘Probabilistic Causality and the Problem of Transitivity,’ Philosophy of Science 50 (1983) 35–57CrossRefGoogle Scholar; Salmon, W. Scientific Explanation and the Causal Structure of the World (Princeton: Princeton University Press 1984)Google Scholar; and Eells.

14 Many people balk at these intuitions; since they are the intuitions of my opponents, I ask the balkers to bear with me.

15 I appeal to Eells, chapter six, in interpreting ‘despite’ as describing negative causation at the singular level.

16 ‘Referring to Events,’ Midwest Studies in Philosophy 2 (1977) 90-9

17 Achinstein, Peter (The Nature of Explanation [Oxford: Oxford University Press 1983], chapter 6)Google Scholar applies modus tollens where Dretske applies modus ponens, concluding that causal contexts are not extensional. Note that the problem described by Dretske is somewhat different from the problem of the contrast class in explanation, which receives its most well-known discussion in chapter five of Bas van Fraassen's The Scientific Image (Oxford: Clarendon Press 1980). In the examples discussed by van Fraassen, the locus of contrast is the explanandum, or effect, rather than the explanans, or cause. For a discussion of the connection between the two problems, see my ‘The Role of Contrast in Causal and Explanatory Claims.’

18 The literature on this topic is vast, and I cannot even scratch the surface here. To my mind the best account of facts and events is Bennett, J. Events and Their Names (Indianapolis: Hackett 1988)Google Scholar.

19 One exception is Lewis's, David theory of events (‘Events,’ in Philosophical Papers, Volume II (Oxford: Oxford University Press 1986) 241-69)Google Scholar; see the discussion in section V below.

20 ‘Causation’ and ‘Postscripts to “Causation”,’ Philosophical Papers, Volume II 159-213

21 ‘Statistics and Causal Inference’

22 Glymour, C. ‘Statistics and Metaphysics,’ Journal of the American Statistical Association 81 (1986), 966Google Scholar

23 Lewis, ‘Postscripts to “Causation”,’ 211Google Scholar

24 ‘A Generalized Probabilistic Theory of Causal Relevance’

25 Noûs 13 (1979) 455-76

26 ‘Counterfactual Dependence and Time's Arrow,’ 472

27 Lewis, ‘Events’

28 ‘Causation,’ 162. In Postscript B, 179-80 Lewis does offer an ingenious account of cases such as Good's where a cause appears to lower the probability of its effect. I have argued elsewhere ('The Mishap at Reichenbach Fall: Singular vs. General Causation’) that this account renders the truth of causal claims unacceptably sensitive to the details of how probabilities evolve with time.

29 As of the time of this writing.