The question I am addressing in this paper is the following: how is it possible to empirically test, or confirm, counterfactuals? After motivating this question in Section 1, I will look at two approaches to counterfactuals, and at how counterfactuals can be empirically tested, or confirmed, if at all, on these accounts in Section 2. I will then digress into the philosophy of probability in Section 3. The reason for this digression is that I want to use the way observable absolute and relative frequencies, two empirical notions, are used to empirically test, or confirm, hypotheses about objective chances, a metaphysical notion, as a role-model. Specifically, I want to use this probabilistic account of the testing of chance hypotheses as a role-model for the account of the testing of counterfactuals, another metaphysical notion, that I will present in Sections 4 to 8. I will conclude by comparing my proposal to one non-probabilistic and one probabilistic alternative in Section 9.
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On the account of testing envisaged by  and sketched above one cannot distinguish between confirmation for a counterfactual of the form ‘if the value of F were set to f, then the value of G would change to g’ and confirmation for a counterfactual of the form ‘if the value of F were set to f, then the probability or chance that the value of G is g would change to 1’.
The Straight Rule, and hence the Axiom of Converge or Reichenbach Axiom, does not enable us to learn hypotheses about objective chances. As is well known, though, it can be reformulated as a rule for inferring limiting relative frequencies. Then one can show that it eventually conjectures values that are arbitrarily close to the limiting relative frequencies in the actual world, provided the latter exist.
I will mostly ignore contexts, as the context sensitivity of counterfactuals does not play a role for the purposes of this paper.
Strictly speaking this notion of admissibility is relative to a context c, just as the notion of admissibility in the Principal Principle is relative to a point of time t. And strictly speaking one needs to assume that the presuppositions of a given context are admissible in this context, and that the world’s theory of deterministic alethic modality, or counterfactuality, is admissible in all contexts. However, the context sensitivity of counterfactuals does not play a role for present purposes, and so the presuppositions of all contexts can be assumed to be tautological. It is perhaps worth noting that history up to some time is a context. For details see (: Section 4).
Spohn (: Chapter 12) proves many much more impressive results that are related to the Obvious Observation. However, the mathematics is not exactly the same, and the interpretation is entirely different. I presently cannot relate his results to mine in an illuminating way.
In general the functions Y i need not have the same index set I as the first family (X i ) i ∈ I . Nor do they have to have a common range 𝓨∗ and associated algebra 𝓥∗. However, this is the special case we are interested in.
Leitgeb’s [28, 29] probabilistic analysis of counterfactuals requires Pr(C ∣ A) = 1 for A ⎕→ C to be true, where Pr is a Popper-Rényi measure ([40, 42]) that is interpreted objectively as time-relative conditional single case chance. Independence in the sense of a Popper-Rényi measure Pr is a relation between three propositions: A is independent of B conditional on C just in case Pr(A∩B ∣ C) = Pr(A ∣ C)⋅ Pr(B ∣ C) (: 103). Counterfactual independence is a relation between two propositions. Therefore the definition of independence in the sense of a Popper-Rényi measure has to be modified so that it becomes a relation between two propositions A and B. It is tempting to say that A is independent of B just in case A is independent of B conditional on the set of all possible worlds, the tautological proposition W. However, on Leitgeb’s logic of counterfactuals, just as on ours, ⊤ ⎕→ α is not logically equivalent to α. Therefore this temptation should be resisted. There are other options for modifying independence in the sense of a Popper-Rényi measure so that it becomes a relation between two propositions (see ). However, without stipulating which modification one chooses there are no formal relations between counterfactual independence, which relates two propositions, and independence in the sense of a Popper-Rényi measure Pr, which relates three propositions. Another complication may arise from a negative answer to the question whether the set of propositions for which time-relative conditional single case chances are defined is as rich as the set of propositions which can figure in the consequent of a counterfactual.
The comparison to  is complicated by the fact that the latter works with a multidimensional possible worlds semantics.
Modal logicians might be interested in the fact that this provides an example of a sentence α such that: ⊩□α and ⊯ α.
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I am very grateful to an anonymous referee, Alan Hájek, Christopher R. Hitchcock, Hannes Leitgeb, Timothy Williamson, and, especially, to Wolfgang Spohn for many most helpful comments and suggestions on several earlier versions of this paper
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Huber, F. What Should I Believe About What Would Have Been the Case?. J Philos Logic 44, 81–110 (2015). https://doi.org/10.1007/s10992-014-9314-x