A principled approach to defining actual causation

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.

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. $\square$