Is time travel (logically) possible? Presumably not in the popular sense, whereby one somehow alters the past. That seems blatantly contradictory. I don't even think we can change the future, let alone the past. We can certainly causally influence the future. But the future itself is never altered. It is different from the present, for sure. But it is never different from itself. All this is made clear on the eternalist conception of a static timeline. Granted, it would be logically possible to influence the past in much the same way as we influence the future. But that would mean that the past was "that way" all along. A time traveler may play a role in making the past what it is. But she couldn't make it what it isn't.
An interesting question arises concerning in what sense the time-traveler actually "travels" back into the past. Such descriptions seem to require Multi-Dimensional Time, as the agent progresses (along her personal timeline) into the historical past. First she is now, and then (later) she is earlier. To avoid contradiction, the 'later' and 'earlier' must be indexed to different temporal axes. It is later in her personal time, but earlier in historical time. If that makes any sense.
But I'm not sure that it really does make much sense to speak of a second temporal dimension, or 'personal time'. More plausibly, the static historical dimension is the only temporal dimension that there is. Suppose S travels far back in time, from t2 to t1. A more careful description might be this: at t1, S pops into existence [call this event A] -- fully formed with "memories" and all -- lives for a while, and then dies. At some other point she is born, lives for a while, then at t2 she pops out of existence [call this event B]. Further, event B causes event A. If B hadn't occurred, then neither would have A.
Excepting minor details, that's the full story of S's time travel, explained with only one temporal dimension. There's no real (metaphysical) sense in which S experiences t1 after t2. It merely seems that way to her because at t1 she has pre-existing memories of t2, whereas at t2 she does not have any memories of t1. We may wish to construct a kind of "representational timeline", i.e. the timeline of her life as she represents it. But that doesn't mean that this other temporal dimension really exists, in any mind-independent sense. It's just a mental construction. (We are inclined to read more into it because we fall under the illusion of endurance, thinking that our momentary self "passes through time" yet retains its absolute numerical identity as it exists wholly in each moment.)
For those who aren't fussed with the eternalist picture, here's a more intuitive argument for the impossibility of changing the past from what it was. (But "what it was" when? Again, we seem to require multiple dimensions, but this time in a plainly incoherent sense. There's t1 as it was originally, without any time travellers. Then there's t1 as it was after the time-traveller went back and "changed" it. But t1 is just t1, the moment in itself, which cannot intrinsically change over time. Before t1 is t0, let us say, and after it is t2. There's no sense to be made of a t1 "before you went and changed the past", as compared to the t1 that exists "after" the time-traveller does their work.)
Oops, lengthy digression. The perils of late-night blogging, hmm. Anyway, my intended argument was this: Suppose (for reductio) that you were to go back in time and change the past. Let's say you assassinate Hitler. Who assassinated Hitler, and why? You -- some guy from the future -- did it, because Hitler wasn't assassinated and that made the world turn out badly. But hang on. Now Hitler was assassinated. So when the future comes around, "you" won't exist. At least, not like that. Not with those memories, those motivations. So the original time-traveller no longer exists, and so cannot do anything at all, let alone assassinate Hitler. So who did assassinate Hitler? Your ghost?
By changing the past, you change the future, and thereby prevent your original time-travelling self (the one with the original past) from ever having existed quite as you had. It's not as obvious as killing your infant parents. But the essential logic is the same. You (with properties P) cause a change, which causes you to not have existed with properties P, and so you-with-properties-P cannot have caused the change after all. Paradox.
(Though note that one way out of this, which I've blogged about before, might be the possibility of branching timelines. On that view, you don't really change the past. You simply change location, i.e. which of the multiple alternative histories you are in. But that would seem to require cross-history causation, and it isn't clear whether that's possible. But maybe it could be made to work. Logically, I mean. We'd just have to treat the multiple spatiotemporal regions as tenuously-connected parts of one big "universe". But then it's more like Sliders than genuine time-travel.)