# Relationship between logical implication arrows and causal diagram arrows

I am reading some of Pearl's more recent work on causal diagrams. It is fun to read but I am struggling to make some connections. Does anyone have intuitive (or precise) knowledge of the relationship between logical implication statements (e.g. $$P(a \Rightarrow b)=P(\neg a \vee b)$$) and the arrows in causal diagrams?

For example, I can estimate the probability above with an empirical relative frequency. If it is close to one I could make a statement like "$$a$$ implies $$b$$ with high probability". Say I do this for a collection of pairs of events. Could I construct a causal diagram(s) from these statements?

• Perhaps you can ask more specifically what you would like to know about the mentioned relationship Mar 25, 2020 at 15:30
• Thanks bee guy. I should go away and do some proper reading on causal discovery algorithms. Hopefully I can come back soon and answer my own question (or delete it).
– Ben
Mar 25, 2020 at 15:51

Usually, the causal diagram arrow relation $$A\to B$$ is interpreted according to the counterfactual statement, "If $$A$$ had not occurred, $$B$$ would not have occurred." See Pearl's The Book of Why, page 265. In logical notation, this is translated as $$\neg A\!\implies\!\neg B,$$ which is equivalent via contrapositive to $$B\!\implies\!A.$$

The most important thing to note here is that causality is fundamentally a counterfactual statement. We're talking about a different universe.

Causation and inference sometimes point in opposite ways: "Your coat is wet because you have come in out of the rain" v. "I know it is raining because your coat is wet".

• Absolutely. I think translating a set of probability statements (empirical or subjective) into a causal diagram is harder than I had hoped though. PC and FCI algorithms might help but I haven't mastered them yet.
– Ben
Mar 25, 2020 at 16:42

I would like it to be otherwise, but it looks like the similar arrow notation for logical implication and in (Pearl's) causal diagrams does not indicate that there is a nice/straightforward connection between the two.

Crucially, it looks like you can't reconstruct a causal diagram with only pairwise statements/observations about co-occurrance of events. There is a nice intro to causal discovery here https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6558187/ which provides many references to related work.

(Although I still hold out some hope that a Causal Inference expert will correct me).