The same logic, i.e., consistency between propositions, is applicable to any narrative whether or not the narrative involves time travel. Thus, if we know that E occurred and if we also know that E could have occurred only if deliberately caused by Bill, then we validly deduce that Bill deliberately caused E. If we ask, "What would have happened if Bill had decided not to cause E?," then the answer is that E would not have occurred.
If E occurred and if we think that it could only have been caused by Bill but are reliably informed that Bill did not cause it, then we can only conclude that we were mistaken to think that only Bill could have caused E. The only impossibility here is the following sequence:
only Bill could have caused E;
Bill did not cause E.
None of this is changed by the additional datum that Bill time traveled from after E to before E in order to cause E.