a forest fire anyway. Similarly, if the lightning had not occurred, there still would have been a forest fire. As we shall see, the HP definition declares both lightning and the arsonist causes of the fire. (In general, there may be more than one actual cause of an outcome.)
A more subtle problem is what philosophers have called *preemption***, which is**
illustrated by the rock-throwing example from the introduction. As **we observed,** according to a naive counterfactual definition of causality, Suzy's **throw would not** be a cause.

The HP definition deals with the first problem by defining causality as counterfactual dependency *under certain contingencies***. In the forest-fire example, the**
forest fire does counterfactually depend on the lightning under the contingency that the arsonist does not drop the match; similarly, the forest fire depends counterfactually on the dropping of the match under the contingency that the **lightning does**
not strike.

Unfortunately, we cannot use this simple solution to treat the case **of preemption. We do not want to make Billy's throw the cause of the bottle shattering by**
considering the contingency that Suzy does not throw. So if our account is to yield the correct verdict in this case, it will be necessary to limit the contingencies that can be considered. The reason that we consider Suzy's throw to be **the cause and** Billy's throw not to be the cause is that Suzy's rock hit the bottle, while **Billy's did** not. Somehow the definition of actual cause must capture this obvious intuition.

With this background, we now give the preliminary version of the HP definition of causality. Although the definition is labeled "preliminary", it is quite close to the final definition, which is given in Section 5. The definition is relative to a causal model (and a context); A may be a cause of B **in one causal model but not in**
another. The definition consists of three clauses. The first and third are quite simple; all the work is going on in the second clause.

The types of events that the HP definition allows as actual causes are ones of the form X1 = x1 ∧ . . . ∧ Xk = xk**—that is, conjunctions of primitive events; this is**
often abbreviated as X~ = ~x**. The events that can be caused are arbitrary Boolean**
combinations of primitive events. The definition does not allow statements of the form "A or A′is a cause of B**", although this could be treated as being equivalent**
to "either A is a cause of B or A′is a cause of B**". On the other hand, statements**
such as "A is a cause of B or B′**" are allowed; this is not equivalent to "either** A is a cause of B or A **is a cause of** B′".

DEFINITION 1. (Actual cause; preliminary version) [Halpern and Pearl 2005] X~ =
~x **is an** *actual cause of* φ in (M, ~u**) if the following three conditions hold:**

## Ac1. (M, ~U) |= (X~ = ~X) And (**M, ~U**) |= Φ.

AC2. There is a partition of V **(the set of endogenous variables) into two subsets**
Z~ and W~ with X~ ⊆ Z~ , and a setting ~x′ and ~w of the variables in X~ and W~ ,