Completing Causal Circles

This post on the circular causality paradox has been occasioned by reading Alan Moore's Jerusalem (London, 2016), BOOK TWO: MANSOUL, Forbidden Worlds, pp. 675-714, but the example has been simplified for discussion purposes.

Assume a single, continuous, immutable timeline:

at time t3, Bill learns of event E that had occurred at t2;

Bill time travels to t1;

between t1 and t2, Bill realizes that E will occur only if he deliberately causes it;

so what happens if Bill decides not to cause E?

The answer is that, if Bill were the kind of person who, in those circumstances, would decide not to cause E, then E would not have occurred and Bill would not, between t1 and t2, be deciding whether to cause it. There is a relevant event in Audrey Niffeneger's The Time Traveler's Wife. Knowing that a picture has been seen undated in the future, the artist decides to date it...but later trims the date off. But I think that I first encountered this aspect of circular causality in a short story by Brian Aldiss.