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]).

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 $p\mathrm{\&}q$-world which is strictly closer to *u* than any $p\mathrm{\&}\mathrm{\neg}q$-world.

Lewis (1986, 79).

Lewis (1986, 79).

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

García-Ramírez (2011, 76).

Which is to say, it satisfies the following properties: if $u{\underset{\xaf}{\prec}}_{w}v$ and $v{\underset{\xaf}{\prec}}_{w}z$, then $u{\underset{\xaf}{\prec}}_{w}z$ (transitivity); $u{\underset{\xaf}{\prec}}_{w}v$ or $v{\underset{\xaf}{\prec}}_{w}u$ (connectedness); and, for all *v*, $w{\underset{\xaf}{\prec}}_{w}v$. It follows in particular that the relation is reflexive.

García-Ramírez (2011, 77).

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, ${w}_{1}$ are related by some or other *degree of closeness* between them” ([2011, 78], my emphasis).

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).

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.

Stevens (1946).

Suppes (1998, 243).

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.

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).

García-Ramírez (2011, 81).

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.

Lewis (1979, 467).

Lewis (1979, 472).

Morreau (2010, 483–90).

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.

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 ${r}_{1}$, whereas *w* is closer than *v* on the other respect ${r}_{2}$. 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 ${r}_{1}$ counts overwhelmingly more than ${r}_{2}$, and a context in which ${r}_{2}$ counts much more than ${r}_{1}$. 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.

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 ${w}_{0}$, there is a $p\mathrm{\&}q$-world ${w}_{1}$ at least as close to *u* as ${w}_{0}$ such that there is no $p\mathrm{\&}\mathrm{\neg}q$-world ${w}_{2}$ at least as close to *u* as ${w}_{1}$. Torza (2012, 25); cf. Burgess (2009, 83 and fol.).

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).

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”.

Williamson (1988, 461).

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.

Williamson (1988, 464).

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.

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).

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

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.

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 ${u}_{C}{u}_{E}$ iff, for every $\mathrm{\neg}C$-world-sum ${v}_{0}{w}_{0}$, there is a $\mathrm{\neg}C\mathrm{\&}\mathrm{\neg}E$-world-sum ${v}_{1}{w}_{1}$ at least as close to ${u}_{C}{u}_{E}$ as ${v}_{0}{w}_{0}$ such that there is no $\mathrm{\neg}C\mathrm{\&}E$-world-sum ${v}_{2}{w}_{2}$ at least as close to ${u}_{C}{u}_{E}$ as ${v}_{1}{w}_{1}$.

Or, for the genuine modal realist, recombination.

Of course, the closest ${v}^{\prime}{w}^{\prime}$ is also such that ${v}^{\prime}$ has exactly the same *K*-particles as *v* and ${w}^{\prime}$ has exactly the same *K*-particles as *w*. The same conditions can be rephrased in terms of counterparts, if necessary.

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

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.

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

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

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

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