Unfortunately, making understanding the real world is not easily reducible to propositional logic. Since you seem interested in pedantry, the technical thing to say is that you must use Bayesian inference and reasonable priors.

In this case, explanations for A and C being correlated without A causing C are strongly related to explanations for why A and B might be correlated without A causing B. (E.g., people who get married are predisposed to be successful.) Evidence, which necessarily can't be in the form of a randomized controlled experiment and yet can still be very strong, that A in fact causes B greatly reduces ones belief that A might not be causing C. My previous comment was asking whether you would defend alternatives that I find to be very unlikely, like a hypothetical predisposition for people to both get married and be happy but which is completely independent from the increase in earnings.