1 Cf. my ‘Counterfactuals and Causal Relevance,’ Pacific Philosophical Quarterly 72 (1991) 314-37, $5, and my A Theory ofCounterfactuals (Indianapolis, IN: Hackett 1986) (henceforth: ATQ, ch. 4, $$I, VI.

2 That the account offered in this paper deals with token, not type, causal relations cannot be overemphasized. Readers are cautioned against contrasting it with much of the literature on probabilistic causation which deals with generic causal relations. But Salmon, d. W. Scientific Explanation and the Causal Order of the World (Princeton, NJ: Princeton University Press 1984)Google Scholar chs. 5, 6, and 7, and Eells, E. Probabilistic Causality (New York: Cambridge University Press 1991)CrossRefGoogle Scholar, ch. 6. Regarding generic causal relations, Humphreys, d. P. The Chances of Explanation (Princeton, NJ: Princeton University Press 1989)Google Scholar, and Cartwright, N. Nature’s Capacities and Their Measurement (New York: Oxford University Press 1989).Google Scholar

3 See my ‘Causal Independence,’ Philosophy of Science 61 (1994) 96-115.

4 See ATC, ch. 2, sections 5, 8, and 9;see also my ‘Counterfactuals,’ Erkenntnis 36 (1992) sections 3, 4, 5, and 6.

5 Given causal relevance, some positive causal impact is interdefinable with purely negative causal impact, and similarly, some negative causal impact is interdefinable with purely positive causal impact. Given causal relevance, there is purely positive causal impact iff there isn’t some negative causal impact, and there is purely negative causal impact iff there isn’t some positive causal impact.

6 I will use the terms ‘being causally relevant’ and ‘having causal impact’ interchangeably.

7 ATC, ch. 4, I.1 and VI; ‘Counterfactuals and Causal Relevance,’ section 5

8 E.g., along the lines of an ordered triple of an object, a predicate (or property) and a time, for a simple case. Alternatively, one might recast such narrow individuation by considering the specified events under their specifying descriptions.

9 This double duty obviates the need for cumbersome locutions, such as ‘the A-event,’ etc. Because the notion of causal impact presented here is a relation between actual events, we will be concerned, when considering the causal impact of A on C, with true sentences A and C.

10 For elaboration of these points, see ‘SPCI.’

11 t_A, the time interval to which A pertains, is the time interval within which the A-event is described as having taken place in the sentence A.

12 (t_A, t_C) is the interval whose starting point coincides with the starting point of t_A, i.e., t_A and whose end point coincides with the starting point of tc, tc.

13 I.e., in (t_A, t_C).

14 See ‘SPCI,’ section 13.

15 Transitivity of some positive causal impact holds under specifiable conditions; cf. ‘SPCI,’ section 13.

16 I.e., clinchers (section VI), where strict reversers are considered not just for aipi and aipd but for (9), (10), and (13) as well (section VII).

17 Or decreaser-clincher (see section VI); i.e., when this is not a two-clincher case.

18 Or increaser-clincher.

19 Thus, no decreasers, hence no strict decreasers.

20 In any year throughout (t_A,t_A,+ 7) and well beyond, his chances (as of t_A) of suffering heart attack due to high blood pressure are lower given A than given ˜A.

21 Assume, for the sake of simplicity, even if it involves some oversimplification, that the function is akin to a normal one. This will also be assumed for the probability functions below. Note that this graph is imprecise: i varies over years, and is thus a discrete parameter. P(D(i)) is a function of i, and assigns a single value for each i. The curves represent P(D(i)) only for discrete values of i, and the values are graphically connected as a heuristic aid.

22 Given the state of the world at t_A, i.e., W_A.

23 That is, there is no decreaser, hence no strict decreaser.

24 That is: it provides the curves for P(D(i)) given very low chance of type 2 pills being taken. Graph 1 reflects the chance at t_A. Given the unlikeliness that type 2 pills will be taken, it can be considered as approximating a case in which type 2 pills are not present. (Of course, the first curve is entirely unaffected by type 2 pills.)

25 This example is a variation of an example suggested by Deborah Rosen, ‘In Defense of Probabilistic Theory of Causality,’ Philosophy of Science 45 (1978) 604-13, discussed by Salmon and Eells.

26 For the sake of simplicity, E was formulated as above. But strictly speaking, as formulated, E is not compatible with ˜A. Compatibility, however, can be secured in the following way. Take E’ as: At t ₁ the vane had an orientation o. Take E” as: At t ₁ the ball (with direction d) hit the vane at spots. Now replace the formulation for E in the text by: E - E’·(˜AVE”). This way, on the left side of the formula for E being an increaser, with A in the condition, the condition (˜A·E·W_A) is tantamount to: A·E’·E”·W_A. E’·E” is tantamount to E in the text. On the right side, the condition (˜A·E·W_A) is tantamount to: E’·-A·W_A. Adding E’ to the condition of P(C/˜A·W_A) doesn’t affect the probability.

27 In A, the exact force in which the ball was hit is not specified.

28 The exact orientation of the vane also depends on the turbulence of the wind with its local variability.

29 Hence P(C/A·W_A) < P(C/˜A·W_A), and aipd obtains.

30 P(C/E·A·W_A) = 0.3 <P(C/E·˜A·W_A) = 0.9.

31 I emphasized in the beginning that opci doesn’t coincide with being a cause. In this paper I explore neither the notion of a cause nor the relation between the two. Notice, though, that whereas in version 1 of this example there is opci and in version 2 there is onci, in both versions A is a cause of C.

32 Even though P(C/E·A.W_A) > P(C/A·W_A), P(C/E·A·W_A) > P(C/E·A·W_A) (= P(C/˜A·W_A)).

33 Rosen’s original example features a branch instead of a vane here. For a discussion of Rosen’scase, see W. Salmon, Scientific Explanation and the Causal Order of the World, 193-5. There (among other differences) the ball hitting the hole replaces the ball crossing the line in our version. Cf. also Eells’s discussion of a variation of Rosen’s example.

34 This is a variation on an example from ‘SPCI.’

35 Assume that the probability of rebellion, given the state of unrest, was very small, and that the conditional dispositions of each senator to favor conciliation given no rebellion were probabilistically in the neighborhood of 1/2.

36 Assume, too, that each was so disposed to the same degree. I assume, as part of the example, that the time interval between the outbreak of the rebellion and the scheduled vote was well beyond that needed for news of the rebellion to reach Rome (and the committee members). Thus, given the rebellion (and W_A), the probability that any senator on the committee was uninformed about the rebellion by the time of the vote was negligible.

37 Assume that, according to the voting procedure, the three senators in tum threw their ballots in isolation from one another (there was no discussion or any interaction between them on the subject). The ballots were then counted by a senate official, and the outcome of the vote formally announced.

38 Mnemonically, E ₂ is associated with two senators, E ₁ with one.

39 The probability of any given senator not voting in accordance with his decision is very low. P(C/˜A· W_A) is about 0.5.

40 I.e.: a decreaser-clincher (in a reversible aipd case, for A and C) is an event F such that;

P(C/F·A·W_A) < P(C/F·˜A·W_A) for which there is no F’ such that P(C/F·F’·A·W_A) > P(C/F·F’·˜A·W_A).

41 Or at least in view of.

42 The subscripts ‘i’ and ‘d’ are used mnemonically.

43 F₁ is a reverser for (9); F ₂ is a reverser for F ₁ and (9).

44 We will assume that P(AIW_A) is about 1/2.

45 Thus, the extents to which the Secretary of Defense was disposed against C — see below (given A) or in favor of C (given ˜A) were about the same (see note 47), and so were the extents to which the Secretary of State was disposed in favor of C (given A) or against C (given ˜A).

46 Note that, as we told the story, the President’s decisions on foreign policy were influenced only by these advisers.

47 Again, the subscripts ‘i’ and ‘d’ are used mnemonically.

48 ˜A·W_A yields, with probability close to 1, that the Belanese government decides in favor of the other option, i.e., to request US forces for combat purposes. Having assumed that P(A/W_A) is about 1/2, and assuming that the strengths of the conditional pre-dispositions of each adviser prior to A are about the same (see above), P(E_i/W_A), P(E_d/W_A) and P(C/W_A) are all about 1/2.

49 The fact that it was a random matter that this occasion happened to be one on which the President asserted her independence and thus acted contrary to the combined weight of her advisers accords with the stricture that A causally impacted the President’s decision only through the influence of her two advisers.

50 The likelihood that the Secretary of State’s influence on the President would drop drastically given F was very high.