of M . Although FL gets as araguments the vale of U, M ,
and FF, in fact, it depends only on the (first component of)
the value of U; that is, FL((i, j), m, f) = i. Similarly, FM ((i, j), l, f) = j. The value of FF depends only on the value of L and M . How it depends on them depends on whether having either lightning or an arsonist suffices for the forest fire, or whether both are necessary. If either one suffices, then FFF ((i, j), l, m) = max(**l, m**), or, perhaps more comprehensibly, FF **= max(**L, M ); if both are needed, then FF **= min(**L, M ). For future reference, call the former model the *disjunctive* model, and the latter the conjunctive model.

The key role of the structural equations is to define what happens in the presence of external interventions. For example, we can explain what happens if the arsonist does not drop the match. In the disjunctive model, there is a forest fire exactly exactly if there is lightning; in the conjunctive model, there is definitely no fire. Setting the value of some variable X to x in a causal model M = (S, F) results in a new causal model denoted MX=x. In the new causal model, since the value of X is set, X is removed from the list of endogenous variables. That means that there is no longer an equation FX defining X. Moreover, X is no longer an argument in the equation FY characterizing another endogenous variable Y . The new equation for Y is the one that results by substituting x for X. More formally, MX=x =
(SX , F
X=x), where SX = (U, **V − {**X}, R|V−{X}) (this notation just says that X is removed from the set of endogenous variables and R is restricted so that its domain is **V − {**X} rather than all of V) and F
X=xassociates with each variable Y **∈ V −{**X} the equation F
X=x Y which is obtained from FY by setting X to x. Thus, if M is the disjunctive causal model for the forest-fire example, then MM=0, the model where the arsonist does not drop the match, has endogenous variables L and FF, where the equation for L
is just as in M, and FF = L. If M is the conjunctive model, then equation for FF becomes instead FF = 0.

In this paper, following HP, I restrict to *acyclic* causal models, where causal influence can be represented by an acyclic Bayesian network. That is, there is no cycle X1, . . . , Xn, X1 of endogenous variables where the value of Xi+1 (as given by FXi+1 ) depends on the value of Xi, for 1 = 1**, . . . , n** − 1, and the value of X1 depends on the value of Xn. If M is an acyclic causal model, then given a context, that is, a setting ~u for the exogenous variables in U,
there is a unique solution for all the equations.

There are many nontrivial decisions to be made when choosing the structural model to describe a given situation.

One significant decision is the set of variables used. As we shall see, the events that can be causes and those that can be caused are expressed in terms of these variables, as are all the intermediate events. The choice of variables essentially determines the "language" of the discussion; new events cannot be created on the fly, so to speak. In our running example, the fact that there is no variable for unattended campfires means that the model does not allow us to consider unattended campfires as a cause of the forest fire.

Once the set of variables is chosen, the next step is to decide which are exogenous and which are endogenous. As I
said earlier, the exogenous variables to some extent encode the background situation that we want to take for granted.

Other implicit background assumptions are encoded in the structural equations themselves. Suppose that we are trying to decide whether a lightning bolt or a match was the cause of the forest fire, and we want to take for granted that there is sufficient oxygen in the air and the wood is dry. We could model the dryness of the wood by an exogenous variable D
with values 0 (the wood is wet) and 1 (the wood is dry).2 By making D exogenous, its value is assumed to be given and out of the control of the modeler. We could also take the amount of oxygen as an exogenous variable (for example, there could be a variable O with two values—0, for insufficient oxygen, and 1, for sufficient oxygen); alternatively, we could choose not to model oxygen explicitly at all. For example, suppose that we have, as before, a random variable M for match lit, and another variable WB for wood burning, with values 0 (it's not) and 1 (it is). The structural equation FWB would describe the dependence of WB on D and M .

By setting FWB (1, **1) = 1**, we are saying that the wood will burn if the match is lit and the wood is dry. Thus, the equation is implicitly modeling our assumption that there is sufficient oxygen for the wood to burn.

According to the definition of causality in Section 3, only endogenous variables can be causes or be caused. Thus, if no variables encode the presence of oxygen, or if it is encoded only in an exogenous variable, then oxygen cannot be a cause of the forest burning. If we were to explicitly model the amount of oxygen in the air (which certainly might be relevant if we were analyzing fires on Mount Everest), then FWB would also take values of O as an argument, and the presence of sufficient oxygen might well be a cause of the wood burning, and hence the forest burning.

It is not always straightforward to decide what the "right" causal model is in a given situation, nor is it always obvious which of two causal models is "better" in some sense. These decisions often lie at the heart of determining actual causality in the real world. Disagreements about causality relationships often boil down to disagreements about the causal model. While the formalism presented here does not provide techniques to settle disputes about which causal model is the right one, at least it provides tools for carefully describing the differences between causal models, so that it should lead to more informed and principled decisions about those choices.

## 3 A Formal Definition Of Actual Cause 3.1 A Language For Describing Causes

To make the definition of actual causality precise, it is helpful to have a formal language for making statements about causality. Given a signature S = (U, V, R), a *primitive* event is a formula of the form X = x, for X ∈ V and x ∈ R(X). A *causal formula (over* S) is one of the form
[Y1 = y1**, . . . , Y**k = yk]ϕ, where 2Of course, in practice, we may want to allow D to have more values, indicating the degree of dryness of the wood, but that level of complexity is unnecessary for the points I am trying to make here.