Abstract
There seem to be two ways of supposing a proposition: supposing “indicatively” that Shakespeare didn’t write Hamlet, it is likely that someone else did; supposing “subjunctively” that Shakespeare hadn’t written Hamlet, it is likely that nobody would have written the play. Let P(B//A) be the probability of B on the subjunctive supposition that A. Is P(B//A) equal to the probability of the corresponding counterfactual, A □→B? I review recent triviality arguments against this hypothesis and argue that they do not succeed. On the other hand, I argue that even if we can equate P(B//A) with P(A □→B), we still need an account of how subjunctive conditional probabilities are related to unconditional probabilities. The triviality arguments reveal that the connection is not as straightforward as one might have hoped.
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Notes
In hindsight, the first detailed investigations into subjunctive supposition took place in debates on (“causal”) decision theory, where the three proposals were discussed e.g. in [24, 33, 34, 36]. (A rare example of an earlier analysis, of a somewhat different form than the ones here discussed, occurs in section 8 of [19]). The idea that debates over the best formulation of causal decision theory can be understood as debates over how to spell out subjunctive conditional probability is emphasized in [14].
In decision theory (see the previous footnote), this corresponds to the proposal in [8, 37]. Note that I use A □→B to stand for a proposition, i.e. a member of the algebra over which the probability measure P is defined. Throughout, I assume that this algebra is atomic, so that propositions can be identified with sets of “possible worlds”, i.e. atoms of the algebra. Some authors defend versions of the Subjunctive or Indicative Equation (see below) in which P(A □→B) or P(A→B) is meant to capture some graded attitude towards a sentence, without assuming that the attitude satisfies the basic rules of the probability calculus. These proposals are outside the scope of the present study.
Neither Lewis nor Kratzer actually explain how these interpretations of (*) and (**) are compositionally derived. If that can’t be done we would have to reconsider the attractiveness of the Lewis-Kratzer account.
Setting aside cases where the supposed proposition A is a contradiction.
On a roughly Lewisian account of subjunctive conditionals, this connection entails a weaker version of the Subjunctive Equation: P(A □→B)≤P (B//A), as shown in [14, 197–199].
One might object that individual worlds should never have positive credence. But the argument also goes through if we replace the worlds with less specific propositions as long as they agree on the chance of B given A and on the truth-value of A □→B.
That kind of situation can easily arise if the chanciness of the coin is the chanciness of statistical mechanics and you happen to have precise control over the microconditions of the toss. See [32] for how the situation can arise even if the coin toss is a fundamental stochastic process and you don’t have a crystal ball.
By clashing with conservativity, the expected chance account not only falsifies the Subjunctive Equation, but also the Subjunctive Inequality mentioned in footnote 6 (and discussed in [38]): if you know A∧B∧C h(B/A)<1, then P(A □→B)>P(B//A). An analogue of conservativity for conditionals is the centring principle A∧B⊧A □→B. If centring holds for conditionals, then by the Subjunctive Equation conservativity must hold for supposition: if P(A∧B)=1 entails P(A □→B)=1, and P(B//A) = P(A □→B), then P(A∧B)=1 must entail P(B//A)=1.
One might be tempted to explain the infelicity by suggesting that the point of subjunctive supposition is to explore genuinely counterfactual possibilities, i.e. possibilities that are known to be false. But that isn’t true. A puzzling event can be explained by pointing out that it would be very likely on the supposition that such-and-such earlier things had happened. This does not presuppose that the earlier things didn’t actually happen. Similarly, in decision contexts an agent may wonder what would happen if she were to choose an option even if she isn’t certain that she won’t actually choose it. We need a concept of subjunctive supposition that allows for cases in which the supposed proposition A has significant positive probability.
The argument generalizes to richer partitions {B,B ′,B ″,…} in place of {B,¬B}.
As Williams notes, if the Subjunctive Equation is replaced by the Subjunctive Inequality from footnote 6, the conclusion (W8) turns into C h(A □→B)≤C h(B). This does not strike me as nearly as problematic as (W8), given the assumptions of the proof. Note that it seems OK if we read A □→B as saying that A nomically necessitates B.
In conversation (2014), Williams suggests that instead of moving to the New Principle, one could stick to the original Principal Principle and modify assumption 1 to say that only the “first-order” restriction of a chance function can evolve by conditionalization, i.e. the part of it that does not concern chance. To make this work, one presumably has to reject the Humean assumption that the “first-order” facts determine the chances. Moreover, once we have exempted chances of chance facts from conditionalization, why couldn’t a friend of the Subjunctive Equation also exempt chances of counterfactuals? In addition, the proposed restriction to assumption 1 arguably does not resolve the more basic worry about applying the simple Principal Principle to propositions about chance: there is simply no good reason to think that chances must be self-aware – especially if we use something like the enrichment technique from Section 2 to get around the fact that actual, physical chance functions may well be undefined for hypotheses about chance.
Moreover, we want \(P_{K_{i}}(B//A)\) to be defined even if \(P_{K_{1}}(A)=0\), in which case \(P_{K_{i}}(B/A)\) may be undefined.
See [31] for an especially transparent implementation of the present strategy.
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Schwarz, W. Subjunctive Conditional Probability. J Philos Logic 47, 47–66 (2018). https://doi.org/10.1007/s10992-016-9416-8
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DOI: https://doi.org/10.1007/s10992-016-9416-8