Causation and Time (Part 2) -- A Time-Ordering Axiom

As we saw in Part 1, Reichenbach used causation to define temporal order by his topological coordinative definition.

(CT) If E₂ is the effect of E₁, then E₂ is called later than E₁.

In this post, I want to recast (CT) as an axiom relating causation and time-order. The usual axioms for causal search -- Markov, Faithfulness, and Sufficiency -- do not mention time. Yet, time-order is often treated as a piece of background knowledge that constrains admissible causal models. Hence, the point of the axiom is to make explicit what assumption under-writes a standard practice in statistics and social science. Recasting (CT), we have:

(T) X → Y only if t(X) < t(Y).

The issue is about under-determination of theory by evidence. Given some data that entail facts about association and conditional association (or independence and conditional independence), we want to build a true model of the causal structure that generated the data. Suppose we know that X, Y, and Z are all pairwise unconditionally associated. Suppose that Y and Z are indpendent conditional on X. And suppose that no other conditional independence relations hold for the data. The usual axioms for causal search yield three models consistent with the data:

(1) Y → X → Z
(2) Y ← X ← Z
(3) Y ← X → Z

Hence, the correct causal model (theory) is under-determined by the data (evidence). Now, suppose that we are told that Y precedes X (i.e. t(Y) < t(X)). Assuming (T), we may eliminate (2) and (3), leaving a unique causal model. Given that Glymour appeals to time-ordering as background information relevant to causal search problems (for example here, see especially page 41), it is somewhat surprising to me that he hasn't already written down a time-ordering axiom. Maybe he thinks it is just too obvious to bother with?

Axiom (T) is intuitive. Saying that it is obvious seems to me to suggest too strongly that it is true; whereas, I don't know whether it is true in general or not. Hence, I think writing down an axiom has two advantages here: (1) it shows explicitly and precisely how the background information matters to the theory, and (2) it lets people prove results with or without the axiom in place.