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Licensed Unlicensed Requires Authentication Published by De Gruyter February 20, 2014

What Trans-World Causation Could and Could Not Be

Alessandro Torza
From the journal Metaphysica


Eduardo García-Ramírez has offered a reductio of the counterfactual analysis of causation. The argument purportedly shows that, given a natural generalization of Lewis’ semantics for counterfactuals, statements expressing the existence of causal dependence across worlds are satisfiable. The aim of the present paper is twofold. In the first part, I show that the purported reductio is flawed, as it relies on an overly strong construal of the semantics for counterfactuals. In particular, it is assumed that we can assign a degree of similarity to any given pair of possible worlds. As it turns out, that assumption reduces to the thesis that the relations of comparative similarity featured in the standard semantics for counterfactuals define an interval scale of measurement on the set of all possible worlds. It will be argued that such a thesis is incompatible with a viable understanding of comparative similarity. The second part of the paper is devoted to a new proof of the possibility of trans-world causation. Nevertheless, the new proof does not amount to a reductio of Lewis’ account of causation per se, but rather of the conjunction of several substantive theses (the counterfactual analysis of causation, modal plenitude, the existence of mereological sums and the best theory account of natural laws).


I am grateful to my colleagues Axel Barceló, Eduardo García-Ramírez and Elias Okon for their careful and extensive comments. I would also like to thank the audiences at Universidad Nacional Autónoma de México, Universidad de los Andes, University of Aberdeen and Universidad Autónoma Metropolitana – Cuajimalpa.


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  1. 1

    Lewis (1973b, 563). Notice that Lewis distinguishes between causation and causal dependence. Since the distinction does not play any role in the present discussion, I will use the two expressions interchangeably throughout, as Lewis does in some contexts (cf. the discussion of trans-world causation in Lewis [1986, 79]).

  2. 2

    Lewis (1973b). Since Lewis rejects the Limit Assumption, technically the expression “the closest p-worlds” may be undefined. The precise truth conditions are as follows. The counterfactual p>q is true at u iff either p is impossible or there is a p&q-world which is strictly closer to u than any p&¬q-world.

  3. 3
  4. 4
  5. 5

    The example is developed in García-Ramírez (2011).

  6. 6
  7. 7

    Which is to say, it satisfies the following properties: if u¯wv and v¯wz, then u¯wz (transitivity); u¯wv or v¯wu (connectedness); and, for all v, w¯wv. It follows in particular that the relation is reflexive.

  8. 8
  9. 9

    García-Ramírez introduces the notion of a degree of closeness in the following passage: “What could the relations that hold between members of pairs of worlds be like? […] Those that Lewis (‘Causation’ and On the Plurality of the Worlds) talks about: relations of comparative similarity among possible worlds […] The actual world, @, and the talking-donkey world, w1 are related by some or other degree of closeness between them” ([2011, 78], my emphasis).

  10. 10

    Hamming distance, a tool of information theory, has been used to measure the distance between epistemic “worlds” by Floridi (2010) among others. Cf. Kracht and Kutz (2006, 956). The method of measuring similarities in terms of Hamming distance has been criticized in Lewis (1973a, 94–5).

  11. 11

    Although in my exposition I used strings of H’s and T’s as representations of worlds, the ersatz modal realist could identify a world with its representation. It follows that the proof of the possibility of trans-world causation is independent of genuine modal realism, which contradicts García-Ramírez’ own claim that his proof of trans-world causation assumes Lewis’ modal metaphysics. Thus, his argument, if valid, is even stronger than originally formulated.

  12. 12
  13. 13
  14. 14

    Scales with an absolute zero, and which are therefore unique up to transformations of the form f(x)=ax, for a>0, are called ratio scales. The Kelvin scale, as well as the mass and weight scales, are all instances of ratio scales.

  15. 15

    Since any two worlds show some kind of similarity (for example, in virtue of both being worlds), there is no such a thing as the zero in a similarity scale. Therefore, a closeness relation cannot define a ratio scale. See Morreau (2010, 485).

  16. 16
  17. 17

    One wrinkle in the example is that the scale has a zero, which occurs when the Hamming distance between two worlds is 10. The model can be easily adjusted so as to avoid defining a ratio scale.

  18. 18
  19. 19
  20. 20

    Morreau (2010, 483–90).

  21. 21

    Kroedel and Huber (2011) discuss an interpretation of the system of weights and priorities of Lewis (1979) in which avoiding “big miracles” (large nomic deviations from actuality) is a dictatorship. Cf. Torza (2012) for an argument that nomic similarity cannot be dictatorial.

  22. 22

    An anonymous referee has suggested that Lewis’ theory of counterfactuals violates Supervenience and, therefore, is immune to Morreau’s result. For instance, suppose there are two respects of comparison such that v is closer than w on one respect r1, whereas w is closer than v on the other respect r2. Now, overall comparative similarity relations are ways of aggregating comparative similarities in particular respects by weighting them on the basis of a certain context. There might then be a context in which r1 counts overwhelmingly more than r2, and a context in which r2 counts much more than r1. Accordingly, there will be an overall similarity ordering on which v is closer, and one on which w is. Therefore, goes the objection, overall similarity does not supervene upon particular similarities. I reply that the appeal to the context-sensitivity of overall similarity is no argument against Supervenience. For the latter tells us that, given a set of world and a set of comparison respects, any function f which maps orderings in particular respects to overall orderings cannot associate two sets of orderings which agree on how they rank v against w in all respects to two overall orderings which disagree on how they rank v against w. This is of course consistent with the existence of distinct mappings f and g which are defined on the same domain of sets of orderings in particular respects but disagree on how they aggregate some particular set of orderings. The difference between f and g may indeed be due to a difference on how they weight each respect of comparison modulo distinct contexts, which captures the intuition underlying the referee’s objection.

  23. 23

    Given a reflexive and transitive relation of closeness to u, the new truth conditions are as follows. The counterfactual p>q is true at u iff, for every p-world w0, there is a p&q-world w1 at least as close to u as w0 such that there is no p&¬q-world w2 at least as close to u as w1. Torza (2012, 25); cf. Burgess (2009, 83 and fol.).

  24. 24

    The possibility of getting the relevant closeness relation by weighing and combining different respects of comparison is what allowed Lewis to respond to Fine’s future similarity objection. See Fine (1975), Lewis (1979).

  25. 25

    Indeed, non-trivial results concerning finite axiomatizability lead Williamson (1988, 472) to suggest that “the four-termed relation is conceptually more basic than its three-termed counterpart”.

  26. 26
  27. 27

    Lewis (1973a, 51). Williamson (1988, 460) has offered his own arguments in support of the symmetry of comparative similarities. Basically, he suggests to add a parameter to the relation T so as to relativize comparisons to a world. When that extra parameter is held fixed, says Williamson, the relation should behave symmetrically. Nevertheless, since the semantics at stake happens not to be defined in terms of closeness relations with the extra parameter, it is not clear why Williamson’s proposal should be relevant to the present issue.

  28. 28
  29. 29

    Lewis (1986, 86) points out that the formulation of plenitude employed here is vacuous for the genuine modal realist, since a way a world could be is a world. For that reason, he provides an alternative construal of plenitude which relies on a recombination principle for worlds – roughly, every way of recombining (duplicates of) individuals constitutes a world. Lewis’ point, however, is orthogonal to the present issue and I will ignore it in the remainder of the discussion.

  30. 30

    A further constraint, which was omitted here, is that the best theory be formulated in a language whose non-logical constants refer only to natural properties. This requirement is meant to avoid achieving simplicity in a trivial fashion. Lewis (1983, 367).

  31. 31

    Lewis defends plenitude in (1986, 86), unrestricted composition in (1986, 213) and the best theory account in (1973a, 73).

  32. 32

    In lieu of referring to the pair of v and w by way of the mereological sum vw, we could have referred to it as a plurality in the sense of Boolos (1984). Therefore, the assumption of unrestricted composition is not essential to the following argument.

  33. 33

    The precise formulation mirrors the truth conditions of footnote 23. So, given a reflexive and transitive relation of closeness, the counterfactual “Had C not occurred, E would not have occurred” is true at uCuE iff, for every ¬C-world-sum v0w0, there is a ¬C&¬E-world-sum v1w1 at least as close to uCuE as v0w0 such that there is no ¬C&E-world-sum v2w2 at least as close to uCuE as v1w1.

  34. 34

    Or, for the genuine modal realist, recombination.

  35. 35

    Of course, the closest vw is also such that v has exactly the same K-particles as v and w has exactly the same K-particles as w. The same conditions can be rephrased in terms of counterparts, if necessary.

  36. 36

    (3) over (4) in Lewis (1979, 472).

  37. 37

    The argument has a wrinkle: it goes through only if the number of particles is finite. It can however be easily generalized to the infinitary case.

  38. 38

    The present objection is based on a comment by an anonymous referee.

  39. 39

    This is the kind of scenario which can take place according to the standard interpretation of quantum mechanics.

  40. 40

    Cf. Okasha (2011) for an application of Arrow’s theorem to the problem of theory choice.

Published Online: 2014-2-20
Published in Print: 2014-4-1

©2014 by Walter de Gruyter Berlin / Boston