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Of Miracles and Interventions

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Abstract

In Making Things Happen, James Woodward influentially combines a causal modeling analysis of actual causation with an interventionist semantics for the counterfactuals encoded in causal models. This leads to circularities, since interventions are defined in terms of both actual causation and interventionist counterfactuals. Circularity can be avoided by instead combining a causal modeling analysis with a semantics along the lines of that given by David Lewis, on which counterfactuals are to be evaluated with respect to worlds in which their antecedents are realized by miracles. I argue, pace Woodward, that causal modeling analyses perform just as well when combined with the Lewisian semantics as when combined with the interventionist semantics. Reductivity therefore remains a reasonable hope.

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Notes

  1. See, for example, Glymour and Wimberly (2007), Halpern (2008), Halpern and Hitchcock (ms.), Halpern and Pearl (2005), Hitchcock (2001a, 2007), Menzies (2004a), Pearl (2009, Ch. 10), and Woodward (2003, pp. 74–86).

  2. See Halpern and Hitchcock (2010, p. 385; ms.), Halpern and Pearl (2005, pp. 846–851), Hitchcock (2001a, pp. 279–284, 2007, pp. 499–504), Menzies (2004a, pp. 821–822), Pearl (2009, pp. 202–215), and Woodward (2003, pp. 42–43).

  3. Woodward thus makes a twofold appeal to causal models. First, he appeals to them at the level of his analysis of actual causation, which is a CMAAC. Second, he appeals to them at the level of his semantics for the counterfactuals encoded in causal models.

  4. At most it tells against the terminological decision to call them ‘counterfactual analyses’ of actual causation (cp. Collins et al. 2004, p. 9).

  5. Thus, although they are standardly called ‘causal’ models, such models don’t encode causal relations, but rather relations of (non-backtracking) counterfactual dependence. (Their alternative name—‘Structural Equations Models’—is less misleading in this regard.)

  6. Terminology due to Halpern and Pearl (2005, p. 844).

  7. For further discussion of model ‘appropriateness’, see Halpern and Hitchcock (2010).

  8. This assumption is implicit in the standard rule—described three paragraphs ago—for evaluating counterfactuals with respect to a model. It is a straightforward consequence of the Lewisian semantics to be described in Sect. 4, below. Woodward (2003, pp. 83–84) also seemingly takes it as true in developing his own CMAAC.

  9. Variants on this second account of permissibility are described by Woodward (2003, pp. 83–84) and by Halpern and Pearl (2005, p. 853). These variants also imply that the actual contingency is always permissible (see Woodward 2003, p. 83; Halpern and Pearl 2005, p. 855).

  10. For related approaches, see Menzies (2004a, b) and Hitchcock (2007).

  11. Note that Woodward is indeed happy to express his interventionist semantics in terms of possible worlds (see ibid., e.g. pp. 139–145).

  12. Simple-antecedent counterfactuals of the form ($) X = x → Y = y and ($$) X = x' □→ Y = y' can be taken as limiting cases of ENFs of form () and (). The truth of such a pair of counterfactuals indicates counterfactual dependence of Y = y upon X = x under the null contingency relative to the direct path 〈X, Y〉 in a model constructed from the variable set {X, Y}, in which there are no off-path variables to hold fixed.

  13. The semantics for counterfactuals that I describe here corresponds to Lewis’s (1979, p. 462) ‘Analysis 1’ of counterfactuals. Lewis attempts to subsume this analysis under a general ‘closest-worlds’ semantics (‘Analysis 2’) that he regards as fully adequate to the ordinary language counterfactual (Lewis 1979, p. 465). For the reason noted at the end of Sect. 1 (the first of the two ‘preliminary points’), this latter project needn’t concern us here.

  14. If \( x^{*} \) = x, so that (*) expresses a true-antecedent counterfactual, no divergence from @ during the transition period starting at \( t_{X = x}^{ - } \) is needed for X = \( x^{*} \) to obtain, and so (assuming determinism) (*) is true just in case y * = y.

  15. As a referee pointed out to me, Loewer (2007, esp. pp. 300–302) observes that a difference in the initial microstates of two worlds may be reflected in a difference in macrostate only much later in time. If this is correct then—where the possible values of X represent macroscopic events or states of affairs—it may be possible to set X to some target value X = x * via a difference in initial microconditions that does not manifest itself macroscopically until at (or shortly before) the time that X comes to take the value X = x *. Hence—where Y is also a macrovariable—this difference in initial conditions may not independently influence Y. It is, however, at best unclear whether implementing counterfactual antecedents in this way will always be possible (especially without an independent influence on Y). In the absence of a proof that it is, I shall follow Woodward and Lewis in supposing that miracles are needed to implement counterfactual antecedents.

  16. More precisely, suppose there is a pair of times, 〈\( t_{{X_{i} = x_{i}^{**} }}^{ = } , \) \( t_{{X_{i} = x_{i}^{**} }}^{ - } \)〉, and an interval of time, \( t_{{X_{i} = x_{i}^{**} }} \), such that \( t_{{X_{i} = x_{i}^{**} }}^{ = } \) is shortly before \( t_{{X_{i} = x_{i}^{**} }}^{ - } \), which is shortly before the beginning of \( t_{{X_{i} = x_{i}^{**}}} , \) and which is also such that the actual laws together with the total state of w* at \( t_{{X_{i} = x_{i}^{**} }}^{ = } \) entail that, throughout \( t_{{X_{i} = x_{i}^{**}}} , \) the event represented by \( X_{i} = x_{i}^{{**}} (x_{i}^{{**}} \ne x_{i}^{*} ) \) will occur. Then w* contains a miracle during the period starting at \( t_{{X_{i} = x_{i}^{**} }}^{ - } \) and ending at the end of \( t_{{X_{i} = x_{i}^{**} }} \) which is just large enough to ensure that X i instead takes value X i  = x * i (as required to implement the antecedent of (**)).

  17. Lewis (1979, p. 463) himself prefers early miracles on the grounds that late miracles make for “abrupt discontinuities” (ibid.). Perhaps it was in part because of worries about the effects of such discontinuities that he rejected late miracles. Still, if Woodward is correct, the early miracles version of the Lewisian semantics doesn't fare any better.

  18. To avoid introducing an undesirable model-relativity into the interventionist semantics, one might demand that I3 be fulfilled with respect to all appropriate models. Alternatively, one might require that it be fulfilled with respect to any model that has a variable set that is ‘sufficiently rich’ in the sense defined by Hitchcock (Hitchcock 2001b, p. 394n): namely (where I L,T is a putative intervention variable for L with respect to T), “[a] variable set would be sufficiently rich if the addition of new variables would not create any new directed paths between variables [I L,T ] and [T], but only interpolate variables along existing paths.”

    For discussion of whether the interventionist semantics renders the truth-values of counterfactuals model-relative and, if so, how this relativity may be overcome, see Strevens (2007, 2008) and Woodward (2008).

  19. It is important here that the non-occurrence of the intervention should be construed as involving the non-occurrence of all of its constituents and not, for example, merely the non-occurrence of the crash-preventing miracle. But this just reflects a requirement that is quite generally essential to the success of CAACs (see Lewis 2004, p. 90).

  20. For simplicity, I’m supposing that the first miracle (and each subsequently introduced miracle) would have at most one effect that threatens to interfere with your exiting. Relaxing this assumption simply means allowing that the intervention I L,T  = i L=0 may have to comprise a still more complex miracle.

  21. The falsity of (CC1) on the Lewisian semantics means only that it delivers the wrong results about causation when combined with the most naïve of CAACs (which take the truth of (CC1) to be necessary for L = 1 to be a cause of T = 1).

  22. It has just been shown that there is counterfactual dependence of T = 1 upon L = 1 under a permissible contingency in the Lewisian model constructed from the variable set {L, C, D 1, D 2,…, D n, T}. Since SCMAACs existentially quantify over models—L = 1 is a cause of T = 1 just in case there is counterfactual dependence of T = 1 upon L = 1 under a permissible contingency in at least one (appropriate) model—it follows that the combination of a SCMAAC with the Lewisian semantics delivers the verdict that L = 1 is a cause of T = 1.

    It might nevertheless be objected that the strategy that I have employed for defending the Lewisian semantics is somehow parasitic upon the interventionist semantics. The objector might say something along the following lines:

    While we can hit upon a set of variables—namely {L, C, D 1, D 2,…, D n, T}—such that there is contingent dependence of T = 1 upon L = 1 in the Lewisian model constructed from that variable set, the Lewisian semantics doesn’t tell us what the relevant variables to include in our variable set are (and hence what to hold fixed in looking for a relation of contingent dependence of T = 1 upon L = 1). The interventionist semantics, by contrast, tells us what the relevant factors to suppress are: namely, those by which a miracle setting L = 0 threatens to independently influence the value of T.

    Such an objector is looking for is a criterion for identifying the variables that must be included in our variable set (if there is to be contingent counterfactual dependence in the Lewisian model constructed from it) which does not simply involve adopting a causal, interventionist-inspired rule along the lines of: “include (and hold fixed by miracles) variables representing those factors by introducing which a miracle setting L = 0 threatens to independently influence T.”

    One plausible criterion for identifying the relevant variables without falling back upon such a rule appeals to the technical notion of a ‘sufficiently rich’ variable set (see fn. 18 above). No Lewisian model with a variable set that doesn't include at least the variables L, C, D 1, D 2,…, D n, and T would count as sufficiently rich (with respect to the variables L and T). The reason is that only in a model that includes all of these variables does the direct path 〈L, T〉 appear (because only by holding fixed the variables C, D 1, D 2,…, D n at their actual values can we recover a counterfactual dependence of T = 1 upon L = 1). The Lewisian can thus say that L = 1 is a cause of T = 1 iff there is contingent counterfactual dependence of T = 1 upon L = 1 in a Lewisian model constructed from a variable set that is ‘sufficiently rich.’

  23. More impoverished variable sets, for which the associated Lewisian model does not contain the path 〈X, Y〉, are not ‘sufficiently rich’ with respect to X and Y.

  24. Where \( t_{L = 1} \) is the time at which you reach the exit, then for any \( t_{L = 1}^{ - } \) (< \( t_{L = 1} \)), if the miracle occurs at \( t_{L = 1}^{ - } \), then those clever scientists will manage to tick box A by \( t_{L = 1}^{ - } + (t_{L = 1} - t_{L = 1}^{ - } )/2. \)

  25. I follow most philosophers in the causal modeling (and wider counterfactual) tradition in assuming that absences (such as the scientists’ not ticking box A) can act as (causes and) effects. It would be straightforward to modify the example so that it involved only positive events.

  26. Taking the relevant miracle to be non-distinct from L = 0 makes no difference to the Lewisian’s response (discussed in the previous section) to the threat of under-generation, which goes through just as before.

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Acknowledgments

For helpful comments, I would like to thank Michael Baumgartner, Christopher Hitchcock, Wolfgang Spohn, two anonymous referees, participants of the ‘Actual Causation Workshop’ in Konstanz in September 2010, and members of the audience for a presentation of an early version of this paper at the British Society for the Philosophy of Science Annual Conference in 2010. This work was supported by the Deutsche Forschungsgemeinschaft (SP279/15-1) and by the James S. McDonnell Foundation Causal Learning Collaborative.

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Glynn, L. Of Miracles and Interventions. Erkenn 78 (Suppl 1), 43–64 (2013). https://doi.org/10.1007/s10670-013-9436-5

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