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disolitude said:

You lose a leg in a boating accident and 10 years later they invent time travel. You travel back in time (one way only) and tell yourself not to go on the boat that caused you to lose your leg. Your past self listens to future self and avoids the boat.

The outcome of this action would be:

1. Your future self would immediatly dissapear as without the boating accident you would never travel back in time to tell past self about the accident. (Looper, Back to the Future time travel ideology)

2. You would not have the accident but your future self would be stuck in the past without a leg for the rest of time. (Terminator time travel ideology)

3. You would lose the leg anyways. Without the boating accident your future self would never go back in time, and since that has to happen you have to lose your leg. (Timemachine time travel ideology)

4. Other - please explain

It depends on your model of time itself. Time could be merely the way that we measure change in the universe, or it could be a fundamental part of its fabric. In the former case, option 2 would be the most likely situation. In the latter case, one would assume that some form of self-correction would have to occur in order to "stabilise" the system... but that could take one of a variety of forms.

The simplest form would in that case be that the future self would change due to the changes in the timeline, and would be seen to travel back in time purely to maintain the timeline - that is, despite not losing the leg, you would remember being visited by your future self and that doing so saved your leg. So you would travel back in time to keep yourself from losing your leg. You could argue that this model was applied, for instance, in Bill & Ted's Excellent Adventure, as well as that Star Trek TNG episode where we see the first time (temporally) that Picard meets Guynan and Data loses his head for a few millennia.

Of course, such a "simple" version only works when there's a logic that can maintain the immediate effects.

But it's likely that the real answer is that time travel to the past isn't possible, due to inherent contradiction. That, or the universe is completely deterministic, so that time travel may happen without creating any sort of loop, in which case any time travel involving the past has already happened. One last possibility is that there's some sort of force that prevents stable loops in time - one that would, for instance, prevent you from killing your grandfather as a child.