Abstract
A number of authors have recently used causal models to develop a promising semantics for non-backtracking counterfactuals. Briggs (Philosophical Studies 160:39–166, 2012) shows that when this semantics is naturally extended to accommodate right-nested counterfactuals, it invalidates modus ponens, and therefore violates weak centering given the standard Lewis/Stalnaker interpretation of the counterfactual in terms of nearness or similarity of worlds. In this paper, I explore the possibility of abandoning the Lewis/Stalnaker interpretation for some alternative that is better suited to accommodate the causal modeling (CM) semantics. I argue that a revision of McGee’s (The Journal of Philosophy 82:462–471, 1985) semantics can accommodate CM semantics without sacrificing weak centering, and that CM semantics can therefore be situated within a general semantics for counterfactuals that is based on the nearness or similarity of worlds.
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Notes
The pass play that Carroll actually did call resulted in an interception that clinched the Patriots’ victory.
See Lewis (1979, p. 472) for the “system of weights or priorities” that is supposed to yield non-backtracking orderings.
Woodward likewise argues (i) that Lewis’s account delivers the wrong non-backtracking verdicts for some counterfactuals, (ii) that Lewis’s similarity criteria are not sufficiently clear to deliver definite truth-values in every case (even once the non-backtracking context has been specified), and (iii) that it is not at all clear why people would or should use Lewis’s criteria to judge the similarity of worlds.
Some readers may seek additional motivation for the exploration of the relationship between the CM approach and the nearest-worlds approach, perhaps because there is no obvious formal reason that we must interpret CM semantics in terms of the nearness of worlds. As I see things, if CM semantics could not be situated within some general semantics that utilizes the nearness of worlds, it would constitute a tentative strike against CM semantics for at least two reasons. First, it seems incumbent upon the defender of CM semantics to produce an intelligible story about how the specific kind of non-backtracking counterfactual reasoning that is captured by CM semantics can be situated within a more general account of counterfactual reasoning. Perhaps that story could be told in some framework that doesn’t deploy the nearness of worlds, but (i) some of the authors of CM semantics (e.g., Pearl 2009, pp. 240–242) have explicitly argued that CM semantics is closely related to the nearest-worlds approach, and (ii) the most influential and (arguably) successful general semantics for counterfactuals all deploy the nearness of worlds—e.g., Kratzer (1981a, b), Lewis (1973a), and Stalnaker (1981). Second, the idea that a counterfactual semantics should in some way track considerations of nearness of worlds is incredibly natural—so much so that it is nearly impossible to explain CM semantics to a non-expert without speaking in terms of the nearest worlds being those that result from intervening. (At least that is my own experience.) Were it possible to preserve the technical soundness of this natural approach while honoring the insights of CM semantics, then this would seem to be a happy result. In this paper, one of my primary aims is to defend the possibility of this happy result against the backdrop of Briggs’s discussion of weak centering and modus ponens.
For dichotomous variables like these, it is customary to allow each variable to take the value of 1 when the event in question obtains, and 0 when it does not obtain.
Whether this result is generated by the CMC depends on one’s preferred axiomatization of the framework. Here, I am following Spirtes, Glymour, and Scheines (2000).
It is not yet settled how CM semantics should be developed to apply to cases where causal relationships are indeterministic, and where the resulting submodel accordingly says that b obtains with some non-extreme probability. Here, I follow Briggs in sectioning off concerns that arise from indeterministic causal relationships by limiting my attention to deterministic causal relationships.
This semantics for counterfactuals is plausible only given Lewis’s (1973b) “Limit Assumption”—i.e., the assumption that “as we proceed to closer and closer a-worlds we eventually hit a limit and can go no farther.” Though Lewis did not always make this assumption, it plays no important role in what follows. So I assume it for ease of exposition. For those who find the Limit Assumption implausible, it is easy to re-describe the worlds that result from intervening to make a happen as closer than any worlds that would not result from intervening to make a happen.
It is likewise not yet clear how one should evaluate counterfactuals with logically complex (e.g., disjunctive) antecedents since it is not clear how one can or should intervene to make logical combinations of states of affairs true. (For example, how should one intervene to make it the case that either Executioner X or Executioner Y did not shoot?) Briggs (2012) extends CM semantics to deal with such cases, but this aspect of the extension is only tangentially relevant to this paper because the treatment of disjunctive antecedents does not yield any obvious tension between general nearest-worlds accounts of counterfactual semantics and CM semantics. That said, as Christopher Hitchcock has pointed out to me in personal communication, there are cases where Briggs’s treatment of disjunctive antecedents may be at odds with the general semantics that I motivate below. This is because there are specific instances (all of which involve disjunctive antecedents) in which import–export is violated by Briggs’s semantics but not by the semantics that I motivate in terms of truth in a world under a set of hypotheses. Since I am not wedded to Briggs’s treatment of disjunctive antecedents here, this doesn’t concern me in the present context. (My reason for setting aside Briggs’s treatment of disjunctive antecedents is that it seems less forced by the standard non-extended of logic of structural equation models than Briggs’s treatment of counterfactual consequents.).
For example, Briggs’s semantics is hyper-intensional in the sense that logically equivalent sentences cannot generally be substituted within the antecedent of a counterfactual.
The antecedent must be true for the simple reason that counterexamples to the validity of modus ponens must be cases where the premises are true (one of which is the antecedent itself) and the conclusion false.
Those who are familiar with Lewis/Stalnaker semantics may intuit that Briggs’s extension is wrong because of their theory-laden tendency to read counterfactuals as claims about the nearest possible worlds in which their antecedents are true. Since these intuitions are no doubt the result of assuming the Lewis/Stalnaker interpretation, they should be set aside in the present context.
Since, as Lewis (1979) points out, we should not confuse this notion of similarity with the common conception of similarity, it may seem that there is no reason to rule out a conception according to which w is not in the set of worlds nearest (or most similar) to itself. The thought would be that the technical conception should not be constrained by this platitude about the common conception. This is wrong for at least two reasons. First, given the Limit Assumption, it is clear that any similarity ordering should be representable as one of Lewis’s (1973a) systems of spheres, and there is no way to draw a system of spheres according to which a point (or w) is not within the sphere in which it resides (i.e., the set of worlds nearest to w). Second, and relatedly, no matter how we understand similarity, it is clear that we should be able to substitute the language of “similarity” with the language of “distance.” Since it is clear that nothing can be nearer or closer to itself than itself, we cannot retain this connection while sacrificing weak centering. (Mathematical characterizations of distance vindicate this point by implying that the minimal distance between any two points is the distance between a point and itself.).
Some authors (e.g., Gundersen 2004) have taken issue with the assumption of weak centering. But in doing so, these authors have typically argued that we should abandon any framework that relies on considerations of nearness or similarity of worlds. Here, when I treat weak centering as self-evident, I mean that it’s self-evident as a claim about the nearness or similarity of worlds, not that it’s self-evident that any counterfactual semantics worth its salt should deploy considerations of nearness or similarity of worlds.
Strictly speaking, McGee shows that any semantics that validates both modus ponens and import–export collapses into the material conditional. But this rules out the possibility of validating both since, as is discussed below, the counterfactual conditional is very clearly not the material conditional.
Because McGee limits his attention to Stalnaker (and not Lewis), he writes as though there must be one uniquely nearest world in which any counterfactual supposition obtains. Once this assumption is dropped (so that multiple worlds can count as equally close), some difficulties emerge. For example, we cannot maintain that ~ p is true under Γ exactly when p is false in some nearest Γ–world. Consider the counterfactual, a > b. Were there to be a tie between nearest a-worlds, and were b to be false in some (but not all) of these worlds, then, given the problematic assumption, the counterfactual, a > ~ b, would qualify as true despite the fact that there would be no sense in which a necessitates ~ b (since some of the nearest a-worlds—indeed, perhaps the great majority—would be b-worlds). This means that we must instead think that ~ p is true under Γ exactly when p is false in every closest world in which every member of Γ is true, and false otherwise. In the special case where Γ is empty—i.e., where we consider whether ~ p is actually true—this may seem problematic, since ~ p could be true of the actual world (and therefore seemingly true) but false in some nearest world (if there is a tie between the actual world and some other world). To avoid this problem, we can assume strong centering—i.e., that every world is uniquely closest to itself—while allowing that worlds that are non-identical to w can be as close to w as each other. (If the actual world is uniquely nearest to itself, and if ~ p is true in the actual world, then ~ p is necessarily true in the nearest world in which the empty set of hypotheses is true.).
This is why row 4 is closer than rows 5, 6, and 7.
This aspect of the semantics may render it of a piece with the premise semantics initially developed by Kratzer (1981a, 1981b), and extended to the causal modeling context by Kaufman (2013) and Santorio (2019). But exactly how tight the connection is requires further exploration, especially since import–export has not been thoroughly investigated in the setting of premise semantics.
In order to develop this semantics in full generality, the mechanics of belief revision must be utilized in order to determine how Γ should be revised when no member of Γ contradicts h, though the set Γ implies ~ h. See, e.g., the AGM model presented in Alchourrón, Gärdenfors, and Makinson (1985).
I am grateful to an anonymous referee for proposing this example.
Some readers may find it intuitive that we should intervene to make executioner X not shoot in order to make the outermost antecedent true, but should not intervene to make executioner Y not shoot in order to make the outermost antecedent true. This is fueled by a reading according to which the outermost antecedent asks us to consider what would happen were executioner X to have done something different than what executioner Y actually did. But on this reading, the outermost antecedent effectively just asks us to consider what would happen had executioner X not shot, while the nested antecedent (according to this kind of reading) just asks us to consider what would happen were executioner Y to shoot. That means that this reading makes the apparent contradiction between the outermost and nested antecedents go away, and there is thus no potential problem for the revised version of McGee’s semantics.
Versions of CM semantics that extend to disjunctive antecedents may stick their necks out in contexts like these, but these extensions do not concern us here (because the standard non-extended logic of structural equation models does not suggest a particular treatment of disjunctive antecedents in the same way that it suggests a particular treatment of counterfactual consequents).
Apart from the revised semantics’ relation to CM semantics, it is worth noting that this is a choice point in the development of the revised semantics. In the event that linguistic intuition disagrees with CM semantics, one could insist that the latter addition of a should not result in the exclusion of ~ a from the set of hypotheses under which b is evaluated. This involves making the revised semantics incompatible with CM semantics (as it stands), but if we strongly intuit that (a & ~ a) > (a > b) and (a & ~ a) > b are not logically equivalent, then it may be worth revisiting CM semantics’ treatment of contradictory antecedents (since there may also be choice points in the development of CM semantics on this front).
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Acknowledgements
For helpful discussion and comments, I am grateful to Malcolm Forster, Daniel Hausman, Christopher Hitchcock, John Mackay, Shanna Slank, Elliott Sober, Michael Titelbaum, and the anonymous referees.
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This article belongs to the topical collection "Conditionals: Truth Conditions, Probability, and Causality", edited by JiJi Zhang, Alan Hajek, and Chin-Mu Yang.
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Stern, R. Interventionist counterfactuals and the nearness of worlds. Synthese 199, 10721–10737 (2021). https://doi.org/10.1007/s11229-021-03265-7
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DOI: https://doi.org/10.1007/s11229-021-03265-7