S is said to be a *witness* for the fact that X~ = ~x is a cause of ϕ according to the causal NESS test.

Unlike the HP definition, causes according to the causal NESS test always consist of single conjuncts.

Theorem 5.3: If {X1 = x1**, . . . , X**k = xk} *is a cause of* ϕ in M *according to the causal NESS test, then* k = 1.

It is easy to check that in Example 3.4, both C1 = 1 and C2 = 1 are causes of WIN = 1 according to the causal NESS test, while (because of NT4) C1 = 1 ∧ C2 = 1 is not.

On the other hand, Example 3.4 shows that neither C1 = 1 nor C2 = 1 is a cause according to the HP definition, while C1 ∧ C2 = 1 is. Thus, the two definitions are incomparable.

Nevertheless, the HP definition and the causal NESS test agree in many cases of interest (in particular, in all the examples in the HP paper). In light of Theorem 5.3, this explains in part why, in so many cases, causes are single conjuncts with the HP definition. In the rest of this section I give conditions under which the NESS test and the HP definition agree. Although they are complicated, they apply in all the standard examples in the literature.

I start with conditions that suffice to show that being a cause with according to the causal NESS test implies being a cause according to the HP definition.

Theorem 5.4: Suppose that X = x *is a cause of* ϕ in (**M, ~u**)
according to the causal NESS test with witness S, and there exists a (possible empty) set T~ *of variables not mentioned in* ϕ or S and a context ~u′*such that the following properties* hold:
SH1. S − {X = x} *is not a sufficient condition for* ϕ in
(**M, ~u**′)*; that is,* (M, ~u′) |= [S − {X = x}]¬ϕ.

SH2. Each variable in T~ is independent of all other variables in contexts ~u and ~u′*; that is, for all variables* T ∈ T~, if W~ *consists of all endogenous variables other* than T , then for all settings t of T and ~w of W~ *, we have*
(**M, ~u**) |= T = t iff (M, ~u) |= [W~ = ~w](T = t), and similarly for context ~u′.

SH3. ϕ is determined by T~ and X in contexts ~u and ~u′*; that* is, for all ~t, T~ ′ disjoint from T~ and X, x
′*, and* ~t
′*, we have*
(M, ~u′) |= [T~ = ~t, T~ ′ = ~t
′, X = x
′]ϕ iff (M, ~u) |= [T~ =
~t, T~′ = ~t
′, X = x
′]ϕ.

SH4. *In context* ~u, S − {X = x} *depends only on* X = x in ~u; that is, for all T~′ disjoint from S and ~t
′*, we have*
(M, ~u) |= [X~ = x, T~′ = ~t
′]S.

Then X = x *is a cause of* ϕ in (**M, ~u**) according to the HP
definition.

Getting conditions sufficient for causality according to the HP definition to imply causality according to the NESS test is not so easy. The problem is the requirement in the NESS
definition that there be a witness S such that (M, ~u′) |= [S]ϕ in all contexts ~u′is very strong, indeed, arguably too strong.

context showing that S − {X = x} is not strongly sufficient for ϕ in NT2, then κ(s~u′ ) ≤ κ(s~u). For ease of exposition, I ignore this issue here.

For example, consider a vote that might be called off if the weather is bad, where the weather is part of the context.

Thus, in a context where the weather is bad, there is no winner, even if some votes have been cast. In the actual context, the weather is fine and A votes for Mr. B, who wins the election. A's vote is a cause of Mr. B's victory in this context, according to the HP definition, but not according to the NESS test, since there is no set S that includes A sufficient to make Mr. B win in all contexts; indeed, there is no cause for Mr. B's victory according to the NESS test (which arguably indicates a problem with the definition).

Since the HP definition just focuses on the actual context, there is no obvious way to conclude from X = x being a cause of ϕ in context ~u a condition holds in all contexts. To deal with this, I weaken the NESS test so that it must hold only with respect to a set U of contexts. More precisely, say that S is sufficient for ϕ *with respect to* U if (M, u) |= [S]ϕ for all u ∈ U. We can then define what it means for S to be *strongly sufficient for* ϕ in (**M, ~u**) *with respect to* U and for X~ = ~x to be a *cause of* ϕ in (**M, ~u**) *with respect to* U in the obvious way; in the latter case, we simply require take strong sufficiency in NT2 and NT3 to be with respect to U.

It is easy to check that Theorem 5.3 holds (with no change in proof) for causality with respect to a set U of contexts; that is, even in this case, a cause must be a single conjunct.

Theorem 5.5: Suppose that X = x *is a cause of* ϕ in (**M, ~u**)
according to the HP definition, with W~ , ~w*, and* x
′ as witnesses. Suppose that there exists a subset W~ ′ ⊆ W~ such that (**M, ~u**′) |= W~ ′ = ~w *(that is, the assignment* W~ ′ = ~w does not change the values of the variables in W~ ′*in context*
(**M, ~u**)) and a context ~u′*such that the following conditions* hold, where W~ ′′ = W~ − W~ ′:
SN1. (M, ~u′) |= [W~ ′ = ~w](X = x
′ ∧ W~ ′′ = ~w).

SN2. W~ ′′ is independent of Z~ given X = x and W~ = ~w in
~u′, so that if Z~ ′ ⊆ Z~, then for all ~z′*, we have* (**M, ~u**′) |=
[X = x, W~ ′ = **~w,Z~** ′ = ~z′](W~ ′′ = ~w).

SN3. ϕ is independent of ~u and ~u′conditional on X and W~ = ~w; that is if Z~ ′ ⊆ Z~, then for all ~z′ and x
′′*, we have*
(M, ~u′) |= [X = x
′′, W~ = ~w
′,Z~ = ~z′]ϕ iff (**M, ~u**) |=
[X = x
′′, W~ = ~w
′,Z~ = ~z′]ϕ.

Then X = x *is a cause of* ϕ in (**M, ~u**) *with respect to* {**~u, ~u**′}
according to the causal NESS test.

6 Discussion It has long been recognized that normality is a key component of causal reasoning. Here I show how it can be incorporated into the HP framework in a straightforward way. The HP approach defines causality relative to a causal model. But we may be interested in whether a causal statement follows from some features of the structural equations and some default statements, without knowing the whole causal model. For example, in a scenario with many variables, it may be infeasible (or there might not be enough information) to provide all the structural equations and a complete ranking function. This suggests it may be of interest to