The statement of the law of logical causality is:
Law of Logical Causality: If conditioning on any event changes the probability an agent assigns to its own action, that event must be treated as causally downstream.
If I'm interpreting things correctly, this is just because anything that's upstream gets screened off, because the agent knows what action it's going to take.
You say that LICDT pays the blackmail in XOR blackmail because it follows this law of logical causality. Is this because, conditioned on the letter being sent, if there is a disaster the agent assigns p=0 to sending money, and if there isn't a disaster the agent assigns p=1 to sending money, so the disaster must be causally downstream of the decision to send money if the agent is to know whether or not it sends money?