Understanding the conditional independence rule in chains Pearl et al. "Causal Inference in Statistics: A Primer" (2016) p. 39 states the following:

Rule 1 (Conditional Independence in Chains) Two variables, $X$ and $Y$, are conditionally
  independent given $Z$, if there is only one unidirectional path between $X$ and $Y$ and $Z$ is any set
  of variables that intercepts that path.

(And then notes that the rule only holds if the error terms associated with the variables are independent of each other.)
As a non-native speaker, I am not entirely sure I understand the rule correctly. Does the if clause say that 


*

*there can only be one path AND 

*that path must be unidirectional? 


Or does it say that


*

*there can be many paths, but 

*among them only a single one is unidirectional? 


Or ...? (My understanding of the English punctuation suggests the second alternative, but my understanding of the context points to the first one.)
 A: I think Pearl is a bit ambiguous. (Thanks to eric_kernfeld for improving my understanding.) From the point-of-view of normal English usage, it is the second understanding. The adjective "unidirectional" modifies the first occurrence of the word "path", making up a single term: unidirectional path. The "if" part says there is only one of those. To say the first, you would have to word it like this:

... if there is only one path, unidirectional, such that...

or

... if there is only one path, that path is unidirectional, and ...

On the other hand, this DAG shows that $X$ and $Y$ can be dependent, even if $Z$ satisfies the second interpretation. Here, $X$ and $Y$ are not independent conditional on $Z,$ even though $Z$ satisfies the second interpretation.

In context, the first interpretation makes more sense.
A: Here's an example that casts doubt on your second interpretation, but is compatible with the first one. Consider the following R code.
a = runif()
b1 = a + runif()
b2 = a + runif()
c1 = b1 + runif()
c2 = b2 + runif()
d = c1 + c2 + runif()

This corresponds to the following DAG. 
a  -> b1 -> c1 
↓           ↓
b2 -> c2 -> d

Suppose we are assessing the independence of A and C1 conditional on {B1,D} as claimed by this lemma or definition. By the second understanding, the criterion is satisfied: there is only one unidirectional path, a->b1->c1, and it is interrupted. By the first understanding, the criterion is not satisfied, because there exists another path a-> b2 -> c2-> d<-c1 (though it is not unidirectional).
Suppose D is 0.1 for this whole example. If C1 is also 0.1, then A must be 0, because $0=D-C_1=C_2\geq A \geq 0$. Under smaller values of C1, A may be as large as 0.1. So in terms of probability theory, A and C1 are not independent conditional on D. Thus, either the lemma describes a concept of conditional independence that is distinct from that typically used in probability theory, or (more likely) the first understanding, not the second, is what Pearl meant -- though you're right that the wording matches the second better.
