This is the second temporal anomaly sometimes known as a Grandfather Paradox.  Last time we considered what happens if a time traveler becomes his own grandfather.  This time we face the paradox in which the time traveler intentionally or unintentionally destroys a link in his own chain of ancestry.

It is interesting that the problem is always phrased with a grandfather.  Rarely does anyone speak of traveling to the past to kill himself, even though that is the practical outcome of the scenario.  This, though, permits fixed time theorists many options in explaining how it is that your efforts did not result in your own undoing.  They would assert first that we know you failed, because had you succeeded you would never have been born to make the attempt.  If pressed--how could you fail if you detonated a nuclear device under his bed while he was in it?--it shifts to whether you killed the right person.  In one way or another, they insist that you cannot kill your true grandfather because history is fixed, and your birth demonstrates that your true grandfather lived to sire your father.

This answer is entirely unsatisfactory for some theorists, who cannot understand how the concerted efforts of untold numbers of time travelers to undo their own lives the easy way could be thwarted by nature, as if it were ommiscient (because it can always predict paradox) and omnipotent (because it can always prevent it).  The fixed time theorist, though, relies on the determinism inherent in that theory, that everything that ever has or ever will happen is already established, and thus whether we travel to the past or stay where we are we cannot really change anything.

For the replacement theory, this is a classic infinity loop scenario:  if the time traveler succeeds in killing his grandfather, he himself will never be born; if he is never born, he will never kill his grandfather; if he never kills his grandfather, he will be born; if he is born, he will kill his grandfather.  While it is hoped that someone attempting to kill his grandfather would fail (which can resolve to an N-jump and the continuation of time), the possibility exists that a free-willed time traveler could succeed in destroying himself and taking the universe with him.  Objections are raised to the infinity loop, and theorists have explored ways to avoid one, but under strict forms of replacement theory it is the unavoidable consequence of the broken causal chain.

Similar broken causal chains occur in many time travel stories.  In Back to the Future the core of the story revolves around the fact that by interfering with his parents' original meeting Marty McFly may have undone his own existence (which would ultimately undo his own interference).  Millennium focuses on the constant danger the time travelers face of destroying their own world.

Next time we will consider temporal doppelgangers or duplicates.