Not necessarily, a variant of TDT could work without logical specification of decisions ...

Would you give a concrete example of what you mean? Right now, I seem to be in the same place where Wei Dai was when he wrote

And you're still using it as a logical fact, i.e., deducing logical consequences from it, right?

I feel like you must be making a point that I'm not getting...

I'm also not seeing the relevance of your reply to him.

When you write:

... it only needs some way of referring to the decision as part of the environment, and tools for figuring out what other facts about the environment follow from the fact that is the decision ...

... in what sense do you understand the "other facts" to "follow from" the "fact that is the decision"? What does the agent work with to determine whether one "fact" "follows from" another?

I agree that the agent needn't be working with strings in some formal language. But, in the decision theories under consideration, the agent is working with some mathematical model of the world, which the agent uses to infer what follows logically from the premise that the agent decides to do X *. I agree also that the agent need not be using a first-order predicate logic to make this inference. Nonetheless, it still seems correct to me to say that what the agent is inferring is a relationship of logical implication.

By analogy, Euclid didn't use a formal first-order predicate logic, but he was still inferring relationships of logical implication.

* I am least familiar with TDT among the decision theories being considered, so, if this statement is wrong, it is most likely wrong about TDT.