Full Idea
It is surely true that if a time traveller travels back in time and succeeds in shooting his youthful grandfather then the time traveller both exists and fails to exist.
Gist of Idea
If a time traveller kills his youthful grandfather, he both exists and fails to exist
Source
Baron,S/Miller,K (Intro to the Philosophy of Time [2019], 8.2)
Book Reference
Baron,S/Miller,K: 'Introduction to the Philosophy of Time' [Polity 2019], p.201
A Reaction
This is the best known paradox of time travel. It is a special dramatic case of making any change to the past. If the traveller kills his neighbour's grandfather, his neighbour should vanish. Moving a speck of dust could have enduring results.
Related Idea
Idea 23021 Lewis said it might just be that travellers to the past can't kill their grandfathers [Lewis, by Baron/Miller]