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Calculus and counterpossibles in science


A mathematical model in science can be formulated as a counterfactual conditional, with the model’s assumptions in the antecedent and its predictions in the consequent. Interestingly, some of these models appear to have assumptions that are metaphysically impossible. Consider models in ecology that use differential equations to track the dynamics of some population of organisms. For the math to work, the model must assume that population size is a continuous quantity, despite that many organisms (e.g., rabbits) are necessarily discrete. This means our counterfactual representation of the model can have an impossible antecedent, giving us a counterpossible. Analogous counterpossibles arise in other sciences, as we’ll see. According to a prominent view in counterfactual semantics, the vacuity thesis, all counterpossibles are vacuously true, that is, true merely because their antecedents are necessarily false. But some counterpossible formulations of differential equation models in science are not all vacuously true—some are non-vacuously true, and some are false. I go on to show how an alternative semantics, one that employs impossible worlds, can deliver this judgment.

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  1. For those conjuring macabre scenarios involving rabbits and blenders, note that I am restricting my attention to living rabbits, since those are what ecologists model.

  2. A related discussion can be found in Weisberg (2012, chapter 4), who claims that differential equations, when applied to discrete organisms, render model systems that can’t be imagined.

  3. Throughout this paper, I’ll be concerned exclusively with deterministic models. This means we won’t need to worry about the various complications that arise when a model predicts the system to be in one among a variety of states at some t. Developing a counterfactual semantics for counterpossibles derived from stochastic models is a project for another day.

  4. Sandgren and Steele (2020) advance a similar argument against general skepticism about the truth of scientific counterfactuals (cf. Hájek 2020).

  5. Sometimes Williamson writes as though vacuists and non-vacuists agree when it comes to sentence (9) (see, e.g., Williamson 2018, p. 364). It’s true that both camps regard the sentence as true, but vacuists think it’s true only because its antecedent is impossible, whereas non-vacuists believe it’s true because of the connection between antecedent and consequent.

  6. This echoes some claims in Williamson (2007), for instance, that there are “reasons to doubt the supposed intuitions” underlying the judgement that counterpossibles can be non-vacuous, which he compares to the “logically unsophisticated” judgment that universal categorical statements with empty subject terms are false (about which more below). But the references to “pre-theoretic” and “pre-reflective” (or “unreflective”) judgments appear to be new.

  7. Thanks to Robert Streiffer for this example.

  8. Ross attempted to do so by arguing that sentences like (12) have a different deep structure than sentences like (11). The merits of Ross’s explanation are controversial (cf. Schmerling 1972).

  9. Williamson also lists an axiom called VACUITY, which says . Despite its name, a non-vacuist can accept VACUITY. For any A, the closest world at which A is false will not be one where A is true. Since is always false, the larger material conditional in which it is an antecedent will always be true.

  10. If one instead prefers to treat worlds as sets of sentences, what I say below would require only minor modifications.

  11. Validity is likewise understood to be truth preservation across all possible worlds, though that will not concern us here.

  12. SD can do all the work that TD does if one assumes that there will automatically be a \(w' \in I\) at which A is true and B is true. This is how Priest approaches the issue.

  13. I have changed Bjerring’s notation slightly.

  14. One nice feature of Bjerring’s formulation is that it’s agnostic about the uniqueness and limit assumptions, about which Stalnaker and Lewis disagreed.

  15. See Nolan (1997, p. 566) for a more precise characterization of SIC.

  16. The way in which I have formulated SSC* does seem in keeping with SIC. According to SIC, no possible world is strange enough to be farther from @ than a world with just one impossibility. It seems to be a natural extension of this condition that the possible (though bizarre) propositions at a world w with just one impossibility will never be enough to outweigh the additional impossible proposition at a w’ with two impossibilities, and hence w’ will be farther from @ than w.

  17. I will assume this objection has nothing to do with a concern about the ontology of impossible worlds. After all, impossible worlds, as construed here, add no new objects to the ontology of one who already countenances sets and propositions (or sentences). Here I’m just discussing the relative complexity of the two theories.

  18. Relevant here is that a model is penalized for its complexity not simply because parsimony is itself treated as a virtue, but because there is a concern that a more complex model will “overfit” the data. It’s unclear whether an analogous concern is even operative in a comparison of Stalnaker–Lewis and the extended version of that theory discussed above, which may undermine using parsimony as a desideratum in the first place.


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I would like to thank Tim Aylsworth, Franz Berto, Hadley Cooney, Peter Godfrey-Smith, Cassandra Grützner, Adam Pham, Graham Priest, Tudor Protopopescu, Reuben Stern, Mike Stuart, Peter Tan, the members of the FraMEPhys reading group (ERC Grant agreement No. 757295)—Michael Hicks, Katie Robertson, and Alastair Wilson–as well as audiences at the 2019 ISHPSSB meeting and the Lingnan University-Hong Kong University summer seminar series for comments and discussion. Two anonymous reviewers for this journal provided quite helpful comments, as well

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McLoone, B. Calculus and counterpossibles in science. Synthese 198, 12153–12174 (2021).

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  • Counterpossibles in science
  • Impossible worlds
  • Idealized models
  • Counterfactual semantics
  • Hyperintensionality