Abstract
In this paper we present a new proposal for defining actual causation, i.e., the problem of deciding if one event caused another. We do so within the popular counterfactual tradition initiated by Lewis, which is characterised by attributing a fundamental role to counterfactual dependence. Unlike the currently prominent definitions, our approach proceeds from the ground up: we start from basic principles, and construct a definition of causation that satisfies them. We define the concepts of counterfactual dependence and production, and put forward principles such that dependence is an unnecessary but sufficient condition for causation, whereas production is an insufficient but necessary condition. The resulting definition of causation is a suitable compromise between dependence and production. Every principle is introduced by means of a paradigmatic example of causation. We illustrate some of the benefits of our approach with two examples that have spelled trouble for other accounts. We make all of this formally precise using structural equations, which we extend with a timing over all events.
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Notes
Surprisingly in the same breath he formulated a different definition as well, known as the regularity account, which is also still influential.
Our formulation and the ensuing principle are not entirely identical to theirs, but the difference is negligible.
For details regarding most of the approaches, again see Weslake (2015).
See Weslake (2015, p. 17) for a discussion.
These examples and this manner of distinguishing between them are due to Lewis (1986).
Here we are using the informal term “prevent” to get across the general idea. The precise interpretation of a timing is given in Definition 10.
Halpern and Hitchcock (2015) provide some of the different views regarding this matter.
The value of \(\textit{BAcc}\) being undetermined can either be interpreted ontologically, meaning there is no fact of the matter what its value would have been had Billy thrown, or epistemically, meaning we simply do not possess any information that establishes the value of \(\textit{BAcc}\). Our approach can be applied using either interpretation.
This difference is not limited to Halpern and Pearl. Collins (2000) and Hitchcock (2001) use the same terminology when discussing which counterfactual scenarios ought to be considered. For example, confronted with Example 7, Collins (2000, p. 8) says that “It is more farfetched, on the other hand, to suppose that the brick wall be absent, or that the ball would miraculously pass straight through it.” Considering an example involving a boulder Hitchcock (2001, p. 298) says of the failure of the backup mechanism that “This possibility is just too farfetched.” Hall and Paul (2003, p. 26) criticise Hitchcock by pointing out the arbitrariness in his use of this terminology.
For details on these situations and the counterexamples they allow, see for example Hall (2007) and Weslake (2015). Halpern (2016) has recently proposed a new definition which is more restrictive, avoiding some of these pitfalls, but not all. Further, it allows for new counterexamples, eg., it fails to judge each of \(\textit{ST}\) and \(\textit{BT}\) a cause in case of SO.
A proof of this theorem is given in the Appendix.
In this respect it is similar to the notion of responsibility as it figures in ethics: ethical judgments concern (for the most part at least) what did happen, not what could have happened. We intend to examine this similarity in more detail in future work.
One should take into account our discussion of early preemption from Sect. 7 though: Weslake uses the deterministic model for EP and still judges there to be causation, whereas we claim there is causation only when using the nondeterministic model.
As a notable exception, Hall’s account (2007) is able to deal with all of these examples succesfully. (Although he would have to add an extra variable to the model for the Backup example, and he disagrees with Weslake on the trumping causation example). Unfortunately it falls victim to other counterexamples, the most wellknown being those from Hitchcock (2009). Again we leave it to the reader to verify that our definition does deliver the right verdict in all of the examples discussed there as well.
An almost identical example is given by Hall (2007), named “backup threat canceller”. He uses it as an example that escapes his earlier dualconcept view of causation as being either dependence or production, and motivated him to develop his later definition. As the analysis will show, our more tolerant notion of production does capture this example. Thus it serves as a good illustration of how our notion of production extends his.
For a very similar example, see “nonexistent threats” (Hall 2007).
References
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Acknowledgments
Sander Beckers was funded by the Flemish Agency for Innovation by Science and Technology (IWT). The authors would like to thank Joe Halpern for interesting discussions on actual causation, as well as two anonymous reviewers for their helpful comments and suggestions on earlier versions of this text.
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Appendix
Appendix
Theorem 1
Given a valid timing \(\tau \), E is dependent on C w.r.t. \((M,\mathbf {u})\) if and only if both of the following conditions hold:

[Condition 1]: C is a producer of E w.r.t. \((M,\mathbf {u},\tau )\).

[Condition 2]: \(\lnot C\) is a producer of \(\lnot E\) w.r.t. \((M_{\textit{do}(\lnot C)},\mathbf {u},\tau _{\textit{do}(\lnot C)})\).
Proof
The implication from right to left is trivial, hence we only need to prove the implication from left to right.
Assume E is dependent on C w.r.t. \((M,\mathbf {u})\), or in other words, \((M,\mathbf {u})\models C \wedge E\) and \((M_{\textit{do}(\lnot C)},\mathbf {u}) \models \lnot E\).
Take \(\tau \) to be any valid timing w.r.t. \((M,\mathbf {u})\), \(n=\tau (E)\), and \(m = \min \limits _{ k \in {\mathbb {N}}} \{L^k_{(M,\mathbf {u})}\text { is} \text {sufficient for } E\}\). We first prove that C is a producer of E w.r.t. \((M,\mathbf {u},\tau )\).
Take \(L^1 \subseteq L^m_{(M,\mathbf {u})}\) to be minimally sufficient for E, i.e., \(L^1\) is sufficient for E, and for any \(L_i \in L^1\), \(L^1 \setminus \{L_i\}\) is not sufficient for E. (Such a set can be constructed by removing elements from \(L^m_{(M,\mathbf {u})}\) one by one.) By construction, all literals in \(L^1\) are direct actual contributors to E. Moreover, since \(m \le n\), these literals are direct producers of E as well.
Since \(\mathbf {U}=\mathbf {u} \subset L_{(M_{\textit{do}(\lnot C)},\mathbf {u})}\), it follows that if \((L^1 \setminus \mathbf {U}=\mathbf {u}) \subseteq L_{(M_{\textit{do}(\lnot C)},\mathbf {u})}\), then \(E \in L_{(M_{\textit{do}(\lnot C)},\mathbf {u})}\), i.e., \((M_{\textit{do}(\lnot C)},\mathbf {u}) \models E\). Therefore there exists at least one endogenous literal \(D \in L^1\) such that \(D \not \in L_{(M_{\textit{do}(\lnot C)},\mathbf {u})}\). By the previous paragraph, D is a direct producer of E.
If \(D=C\), then we are finished with this part of the proof. So assume \(D \ne C\). We can apply the exact same reasoning as we did for E, to find a direct producer F of D such that \(F \not \in L_{(M_{\textit{do}(\lnot C)},\mathbf {u})}\). Since production is transitive, F is a producer of E as well. Given that there are only a finite number of endogenous literals, and that M is assumed to be acyclical, continuing this reasoning will eventually end up with finding C as a producer of E. Therefore we conclude that C is a producer of E w.r.t. \((M,\mathbf {u},\tau )\).
We can apply the exact same reasoning to prove that also \(\lnot C\) is a producer of \(\lnot E\) w.r.t. \((M_{\textit{do}(\lnot C)},\mathbf {u},\tau _{\textit{do}(\lnot C)})\), which concludes the proof. \(\square \)
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Beckers, S., Vennekens, J. A principled approach to defining actual causation. Synthese 195, 835–862 (2018). https://doi.org/10.1007/s1122901612471
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DOI: https://doi.org/10.1007/s1122901612471