As someone who loves a good time travel story (note my avatar and my signature
) if classic time travel is indeed possible, then it depends on the rules of the game.
In Connie Willis' book
To Say Nothing Of The Dog, the catch is that the Universe just doesn't allow paradoxes. If you tried to go back and kill an ancestor, you wouldn't be able to accomplish it. Despite your best efforts, you might miss your shot, chicken out, "land" too far away or at the wrong time to get the job done etc, etc... All of your actions would result in the same old present. The movie
12 Monkeys goes the same route.
In multiple universe stories, such as
The Man Who Folded Himself you can't ever go back to your original timeline so the Grandfather you kill is a different man in many ways. So are you.
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Originally Posted by Nestoeman
time travel should'nt be possible due to conservation of matter (IMO) because the past has no space for addition (matter/energy), and creating multiply dimensions can't happen cus of this either.
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I believe this is the greatest hurdle to the notion of classic time travel. It's fun to think of the big paradoxes, but the mere existence of your molecules occupying the area where other molecules are is a paradox in itself. Perhaps this is allowed on the quantum level, but it suggests to me that the Grandfather paradox is Moot.
My favourite story:
The Light of Other Days by Arthur C. Clarke and Stephen Baxter has a nifty way around this, but still doesn't allow for paradoxes.