Abstract
I present and discuss two logical results. The first shows that a non-trivial counterfactual analysis exists for any contingent proposition that is false in at least two possible worlds. The second result identifies a set of conditions that are individually necessary and jointly sufficient for the success of a counterfactual analysis. I use these results to shed light on the question whether disposition ascribing propositions can be analyzed as Stalnaker-Lewis conditional propositions. The answer is that they can, but, in order for a counterfactual analysis to work, the antecedent and consequent must be related in a particular way, and David Lewis’s Time’s Arrow constraints on comparative world similarity must be relaxed. The upshot is that counterfactual analyses are easy to come by, in principle, even if not in practice. In that sense, it’s easy to be iffy.
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Notes
Some authors describe finks as a species of mimick and reverse finks as a species of mask, but it will facilitate my later discussion to follow Johnston (1992) in restricting the labels ‘mask’ and ‘mimick’ to cases that do not involve actual or hypothetical alterations in a thing’s dispositional properties.
Bird (2007) refers to masks as ‘antidotes’, which is the more natural term in the context of examples like the disposition of a poison to cause death when ingested.
There are, of course, a number of more finegrained ways of identifying and individuating propositions. Even under a more finegrained conception of propositions, the result proved below will show that every proposition is necessarily equivalent to a counterfactual. Such a necessary equivalence suffices to constitute a counterfactual analysis in any case.
In Section 7, below, for consistency with another author’s notation, I will write ¬A in place of \(\overline {A}\) and A ∧ B in place of AB.
In this respect, the First Conditional Analysis Theorem contrasts with Holton’s (1999) construction of a four-world model in which four contingent propositions are defined in terms of one another as counterfactuals with respect to a single measure of comparative world similarity. Holton’s express aim there is to argue for the logical coherence of the thesis that all truths are dispositional, but in order for every proposition to be defined as a counterfactual, his setup, unlike ours, requires that not every set of possible worlds count as a proposition.
Note that the generated propositions are not “new”. Each is a set of possible worlds that exists regardless of any counterfactual operator.
I thank Branden Fitelson for pointing out to me that this proposition is always an option.
Note that choosing A and B so that AX entails B and AB entails X does not guarantee that X = (A > X).
This is because, in Stalnaker-Lewis counterfactual logic, the propositions A > B and A > (A > B) are identical, so if X = (A > B), then X = (A > B)=(A > (A > B))=(A > X). The importance of the equivalence of A > B and A > (A > B) for the tenability of counterfactual analyses of dispositions is emphasized by Bonevac et al. (2006).
For a discussion of a different sort of case in which the counterfactual operator needs a non-standard resolution of vagueness, see (Cross forthcoming).
This is guaranteed by Lewis’s (1973, p. 48) condition (3).
Stalnaker’s and Lewis’s preferred systems both validate the principle of conditional non-contradiction: ◇p, p > ¬q ⊩¬(p > q).
This argument incorporates the reasoning represented in (Bonevac et al. 2006, Fig. 1, p. 285), except that Bonevac et al. omit (2) as a member of Σ C . I include (2) in order to apply conditional non-contradiction from Stalnaker-Lewis conditional logic. Bonevac et al. instead apply a stronger principle, which they call “Exclusion”, at step 10 of their Fig. 1.
It might be objected that Lewis was trying to capture not only a range of ordinary contexts but also something about the metaphysical structure of reality, namely the temporal asymmetries of counterfactual dependence. This objection misses the mark, however, because, for Lewis, the temporal asymmetries of counterfactual dependence are not part of the metaphysical structure of reality. They are not even exceptionless: “I do not claim that the asymmetry holds in all possible, or even all actual, cases. It holds for the sorts of familiar cases that arise in everyday life. But it well might break down in the different conditions that might obtain in a time machine, or at the edge of a black hole, or before the Big Bang, or after the Heat Death, or at a possible world consisting of one solitary atom in the void. It may also break down with respect to the immediate past. (Lewis 1979, p. 458)”
I owe this example to Don Nute.
That is, Lewis’s Time’s Arrow constraints make no logical difference in the context of determinism. In the context of indeterminism, much can be said about how past and future differences differently affect comparative world similarity, and in the context of indeterminism, these different effects on comparative world similarity do make a logical difference, as Thomason and Gupta (1980) and Nute (1991) show.
We still face the Duhem-Quine problem, of course. Maybe the truth of A C was not really brought about; maybe the cup didn’t really break.
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Acknowledgements
This research was supported in part by two grants from the University of Georgia: an M. G. Michael Award and a Provost’s Summer Research Grant. For their comments on previous versions, I am grateful to Tony Dardis, Branden Fitelson, Donald Nute, the anonymous referees, and to audiences at the Munich Center for Mathematical Philosophy and the 2015 meeting of the Society for Exact Philosophy at McMaster University.
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Cross, C.B. Every Proposition is a Counterfactual. Acta Anal 31, 117–137 (2016). https://doi.org/10.1007/s12136-015-0268-6
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DOI: https://doi.org/10.1007/s12136-015-0268-6