In my previous post I tried to get clear about when variables could be safely removed from a causal model without affecting what the model is capable of telling us about singular causal relations. There, I endorsed two principles stating when causal models may be *reduced* by excising variables in a particular way. If we endorse these principles, and we want to give a theory of singular causation formulated in terms of correct causal models, then we should want that theory to give the very same verdicts before and after model reduction. The point of today’s post is that there is a wide family of theories of causation which run afoul of this constraint. Those theories will say that two variable values are causally related in one model, but reverse this judgment when the model is reduced.

### 1. Counterfactual Counterfactual Theories of Singular Causation

#### 1.1 Counterfactuals in Causal Models

Causal models allow us to evaluate certain causal counterfactual conditionals. For instance, recall the causal model describing the relations of causal determination between whether the switch is up, whether the power is on, and whether the light is illuminated.

$$
\begin{aligned}
L &:= S \wedge P \\\

P &:= S
\end{aligned}
$$
Suppose that, actually, the switch is down, $S=1$, so that the power is on and the light is illuminated. If we want to evaluate the counterfactual conditional $P = 0 \hspace{4pt}\Box\hspace{-4pt}\to L = 1$ (were the power off, the light would be illuminated), we *mutilate* the model $\mathbb{M}$ by removing $P$'s equation, severing $P$'s dependence upon $S$, and setting its value to $0$ directly. That is, we *exogenize* the variable $P$, and add the assignment $P=0$ to the context $\vec{u}$. Graphically, we cut the arrow going into $P$, but leave all other arrows intact.

Call the resulting mutilated model “$\mathbb{M}[P \to 0]$”. The semantics for counterfactuals tells us that $P = 0 \hspace{4pt}\Box\hspace{-4pt}\to L = 1$ is true in the model $\mathbb{M}$ iff $L=1$ is true in the mutilated model $\mathbb{M}[P \to 1]$. $$ \mathbb{M} \models P = 0 \hspace{4pt}\Box\hspace{-4pt}\to L = 1 \quad \iff \quad \mathbb{M}[P \to 0] \models L =1 $$ Since $L=0$ in the mutilated model $\mathbb{M}[P \to 0]$, this tells us that the counterfactual $P = 0 \hspace{4pt}\Box\hspace{-4pt}\to L = 1$ is false in the original model $\mathbb{M}$.

More generally, if $\vec{X}$ is a vector of variables in $\mathbb{U} \cup \mathbb{V}$ and $\vec{x}$ is some assignment of values to those variables, then we may define $\mathbb{M}[\vec{X} \to \vec{x}]$ to be the mutilated model that you get by going through each variable $ X \in \vec{X}$ and, if $X$ is endogenous, removing $X$'s structural equation $\phi_X$ from $\mathbb{E}$, moving $X$ from $\mathbb{V}$ to $\mathbb{U}$, and adding the assignment $\vec{x}(X)$ to the context $\vec{u}$. (By the way, “$\vec{x}(X)$” is the value which $\vec{x}$ assigns to the variable $X$.) On the other hand, if $X \in \vec{X}$ is exogenous, then you simply change the context so that $\vec{u}(X) = \vec{x}(X)$. Then, for any $\phi$, we have that $$ \mathbb{M} \models \vec{X} = \vec{x} \hspace{4pt}\Box\hspace{-4pt}\to \phi \quad\iff \quad \mathbb{M}[\vec{X} \to \vec{x}] \models \phi $$

#### 1.2 Counterfactual Counterfactual Depdendence

Many contemporary theories of causation fit into the following general schema, which we can call “**Counterfactual Counterfactual**":

**Counterfactual Counterfactual**. $C=c$ caused $E=e$ in causal model $\mathbb{M}$ iff there is some value of $C$, $c'$, such that
$$
\mathbb{M}[\vec{G}\to\vec{g}] \models C = c’ \hspace{4pt}\Box\hspace{-4pt}\to E \neq e
$$
for some suitable vector of variable $\vec{G}$ and a suitable assignment of values $\vec{g}$.

According to **Counterfactual Counterfactual**, causation is not counterfactual dependence; rather, it is counterfactual dependence in some counterfactual scenario, $\vec{G} = \vec{g}$. Assuming that the empty vector of variables counts as suitable, **Counterfactual Counterfactual** will entail that counterfactual dependence is sufficient for causation.

We will get different theories of causation depending upon which vectors of variables, and which assignments of values, we take to be *suitable*. For instance, the account of Hitchcock (2001) tells us that $\vec{G}$ and $\vec{g}$ are suitable iff, in $\mathbb{M}$, there is some directed path leading from the variable $C$ to the variable $E$, $C \to V_1 \to V_2 \to \dots \to V_N \to E$, such that, in the counterfactual model $\mathbb{M}[\vec{G} \to \vec{g}]$, every variable $V$ along this path retains its actual value in the original model, $\vec{u}(V)$.

**Hitchcock (2001)**. $\vec{G}$ and $\vec{g}$ are suitable iff there some some path from $C$ to $E$ such that, for every variable $V$ along this path,
$$
\mathbb{M}[\vec{G} \to \vec{g}] \models V = \vec{u}(V)
$$

There are some cases in which Hitchcock looks too strong (e.g., the Voting Machine case from appendix A.2 of Halpern & Pearl (2005)). These and other cases were taken to motivate a move to the following weaking.

**Halpern and Pearl (2005)**. $\vec{G}$ and $\vec{g}$ are suitable iff, for all vectors of variables $\vec{P}$ not in $\vec{G}$, and any subvector $\vec{H}$ of $\vec{G}$,
$$
\mathbb{M}[\vec{H} \to \vec{g}(\vec{H}), \vec{P} \to \vec{u}(\vec{P}), C \to c] \models E=e
$$

Notice that, if a counterfactual setting $\vec{G} = \vec{g}$ is suitable according to Hitchcock (2001), then it will automatically be suitable according to Halpern and Pearl (2005). So, if $C=c$ caused $E=e$ according to Hithcock (2001), then $C=c$ caused $E=e$ according to Halpern and Pearl (2005).

Both of these accounts of causation face a problem with cases of what’s come to be known as *bogus prevention*, illustrated by the neuron diagram below.

In this neuron diagram, $C$'s firing does not prevent $E$ from firing (that is: $C$'s firing did not cause $E$ to not fire). However, both Hitchcock (2001) and Halpern and Pearl (2005) get the verdict that $C$'s firing prevented $E$ from firing. That’s because both of them rule the singleton vector of variables $\vec{G} = (A)$, with the assignment $\vec{g}=1$, suitable. And, in this counterfactual setting, whether $E=0$ counterfactually depends upon whether $C = 1$.

In response to cases like these, there has been further emendation of the Halpern and Pearl account to incorporate standards of *normality*, or *typicality*. Halpern (2008) emends the Halpern and Pearl (2005) account like so:

**Halpern (2008)**. $\vec{G}$ and $\vec{g}$ are suitable iff, for all vectors of variables $\vec{P}$ not in $\vec{G}$, and any subvector $\vec{H}$ of $\vec{G}$,
$$
\mathbb{M}[\vec{H} \to \vec{g}(\vec{H}), \vec{P} \to \vec{u}(\vec{P}), C \to c] \models E=e
$$
and, in addition, there is some assignment of values to the variables in the model such that, in that assignment, $\vec{G} = \vec{g}$ and $C = c'$, and that assignment is *at least as normal, or typical* as the variable assignment of the original model $\mathbb{M}$.

This definition requires us to outfit our causal models with a ranking over assignments of values to all of the variables in $\mathbb{U} \cup \mathbb{V}$. There will be complicated questions about which variable values are more normal than which others; however, it we restrict our attention to simple neuron diagrams, we can at least rest assured that everybody seems to agree that it is more normal or typical for a neuron to *not* fire than it is for it to fire. If we assume that $A$'s *not* firing is more normal that $A$'s firing, then Halpern (2008) tells us that the counterfactual setting $A=1$ in *Bogus Prevention* is not suitable; and, therefore, that $C=1$ did not cause $E=1$.

Notice that, if a counterfactual setting $\vec{G} = \vec{g}$ is suitable according to Halpern (2008), then it will automatically be suitable according to Halpern and Pearl (2005). So, if $C=c$ caused $E=e$ according to Halpern (2008), then $C=c$ caused $E=e$ according to Halpern and Pearl (2008).

For further discussion of these accounts, see chapters 7 and 8 of my Seminar Notes for *Causality*

### 2.Counterfactual Counterfactual Accounts Reverse Causal Judgments in Model Reductions

Recall the Lewisian neuron diagram of a case of preemption.

We may model this neuron diagram with the following system of structural equations (where the variables have the natural interpretation, with $1$ corresponding to firing and $0$ corresponding to not firing):
$$
\begin{aligned}
E &:= B \vee D \\\

D &:= C \\\

B &:= A \wedge \neg C
\end{aligned}
$$
(The context is just $C=1$ and $A=1$.) Let’s call this model “$\mathbb{M}$”. In the original neuron diagram, $C$'s firing is a cause of $E$'s firing. So we should want our theory of singular causation to tell us that, in this causal model, $C=1$ is a cause of $E=1$. Getting cases like this right is non-negotiable for a theory of singular causation. And, fortunately, counterfactual counterfactual accounts like Hitchcock’s (2001) and Halpern and Pearl’s (2005) are capable of saying that $C=1$ is a cause of $E=1$ in the causal model above. To deliver this verdict, those theories let $\vec{G} = (B)$, with $\vec{g} = (0)$. As you may verify for yourself, Hitchcock (2001), Halpern and Pearl (2005), and Halpern (2008) all deem this choice suitable. But then,
$$
\mathbb{M}[B \to 0] \models C = 0 \hspace{4pt}\Box\hspace{-4pt}\to E = 0
$$
That is: in the counterfactual scenario where $B$'s value is held fixed at $0$, had $C$ not fired, $E$ would not have fired either. So, **Counterfactual Counterfactual** deems $C=1$ a cause of $E=1$.

Note that the following is an exogenous reduction of this model in which we have excised the exogenous variable $A$ by substituting $1$ for $A$ in $B$'s structural equation.
$$
\begin{aligned}
E &:= B \vee D \\\

D &:= C \\\

B &:= \neg C
\end{aligned}
$$
Call the resulting model “$\mathbb{M}_A$”. The endogenous variable set of $\mathbb{M}_A$ is non-empty and the equation for $B$ is still surjective, so this is a valid exogenous reduction. By our principle **Valid Exogenous Reduction Preserves Correctness** (see the previous post), $\mathbb{M}_A$ is correct if the original model $\mathbb{M}$ was.

Given $\mathbb{M}_A$, we may excise the endogenous variable $B$ by removing $B$'s structural equation and substituting $\neg C$ for $B$ in $E$'s structural equation.
$$
\begin{aligned}
E &:= \neg C \vee D \\\

D &:= C

\end{aligned}
$$
Call the resulting model “${\mathbb{M}_A}_B$”. $B$ is not a collider in $\mathbb{M}_A$, so this is a valid endogenous reduction of $\mathbb{M}_A$. By our principle **Valid Endogenous Reduction Preserves Correctness** (see the previous post), ${\mathbb{M}_A}_B$ is correct if $\mathbb{M}_A$ was.

However, just considering *this* model, **Counterfactual Counterfactual** tells us that $C=1$ is *not* cause of $E=1$. For the only possible choices of $\vec{G}$ are the empty vector and the singleton vector $(D)$. Since $E=1$ does not counterfactually depend upon $C=1$, the empty vector does not witness $C=1$'s causing $E=1$. And
$$
{\mathbb{M}_A}_B[D \to 1] \models C=0 \hspace{4pt}\Box\hspace{-4pt}\to E=1
$$
So there is no counterfactual dependence between $E=1$ and $C=1$ in the counterfactual scenario in which $D$ is held fixed at $1$. And
$$
{\mathbb{M}_A}_B[D \to 0] \models C=0 \hspace{4pt}\Box\hspace{-4pt}\to E=1
$$
So there is no counterfactual dependence between $E=1$ and $C=1$ in the counterfactual scenario in which $D$ is held fixed at $0$. So there is no counterfactual dependence between $E=1$ and $C=1$ period. So they are not causally related, according to **Counterfactual Counterfactual**.

What we’ve just seen is that, if we accept the principles on valid model reduction from the previous post, then the verdicts of a theory like **Counterfactual Counterfactual** vary from correct model to correct model. Above, we relied upon both the principle **Valid Exogenous Reduction Preserves Correctness** and **Valid Endogenous Reduction Preserves Correctness**. However, we can get Halpern (2008) to flip its verdict by just excising an exogenous variable from a correct causal model.

Consider the following neuron diagram.

Here’s how to read this diagram: if $B$ fires at $t_1$, then it will cancel out any *one* signal sent from $A$ or $C$. So, if $B$ fires and exactly one of $A$ and $C$ fire, then $E$ will not fire. If $B$ fires and both $A$ and $C$ fire, then $E$ will fire. And, if $B$ doesn’t fire, then $E$ will fire iff at least one of $A$ and $C$ fire.

We can represent this neuron diagram with the following structural equation. $$ E := (\neg B \wedge (A \vee C)) \vee (B \wedge (A \wedge C)) $$ (The context is $A=C=1$ and $B=0$.) Call the causal model containing these variables, this context, and this equation “$\mathbb{M}$”. Given given the natural assumption that not firing is more normal, or typical, than firing, Halpern (2008) tells us that $C$'s firing ($C=1$) is a cause of $E$'s firing ($E=1$) in $\mathbb{M}$. That’s because the variable assignment in which none of the neurons fire is more normal that the actual variable assignment, and this is an assignment in which $A=C=0$. So, Halpern (2008) tells us that the counterfactual setting $A=0$ is suitable; and, in this counterfactual setting, whether $E=1$ counterfactually depends upon whether $C=1$. $$ \mathbb{M}[A=0] \models C = 0 \hspace{4pt}\Box\hspace{-4pt}\to E = 0 $$

So, $C=1$ caused $E=1$. Now, I don’t think that this verdict is a desideratum of a theory of causation. I, like Lewis and Mackie, am content with an account which says that, while neither $A=1$ nor $C=1$ individually caused $E=1$, the *disjunction* (or the fusion, or what-have-you) of $A=1$ and $C=1$ did. However, I am also content with an account according to which $C=1$ was a cause of $E=1$. And that is how Halpern (2008), like Hitchcock (2001) and Halpern and Pearl (2005), comes down on this case.

Suppose that we excise the exogenous variable $A$ from this model. This gives us the new model $\mathbb{M}_A$, which contains the variables $C, B,$ and $E$, the structural equation
$$
E := \neg B \vee C
$$
The resulting endogenous variable set is non-empty, and the resulting structural equation is surjective, so this exogenous reduction is valid. By our principle **Valid Exogenous Reduction Preserves Correctness**, $\mathbb{M}_A$ is correct if $\mathbb{M}$ was.

Now, while Halpern (2008) said that $C=1$ caused $E=1$ in $\mathbb{M}$, it reverses this judgment in $\mathbb{M}_A$. That’s because, in the actual context $B=0$, whether $E=1$ does not counterfactually dependend upon whether $C=1$. And while, in the counterfactual setting $B=1$, whether $E=1$ does counterfactually depend upon whether $C=1$,
$$
\mathbb{M}_A[B \to 1] \models C=0 \hspace{4pt}\Box\hspace{-4pt}\to E=0
$$
This counterfactual setting is not suitable, according to Halpern (2008). For having $B$ *fire* is less normal than having $B$ not fire. (Or, if we reject this normality ranking, for whatever reason, this calls into question whether the account is capable of getting the right verdict in *Bogus Prevention*.)

So, again, if we accept the principle that **Valid Exogenous Reduction Preserves Correctness**, then the verdicts of Halpern (2008) vary from correct model to correct model.