The Grandfather Paradox
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Tim hates Grandfather. Tim hates him so much that his ambition is to murder Grandfather despite the fact that Grandfather is already dead: he died in his sleep in 1957. Tim is no quitter, however, so he builds a time machine and travels back to the year 1920, a time before Grandfather's death. Tim buys a high-powered rifle, practices his marksmanship for many days, rents a room along the path Grandfather takes to work every day, and waits for optimal conditions. When the time is right, Tim locks and barricades the door to avoid any intrusion, and otherwise prevents any factors that might keep him from hitting his target. Tim is perfectly accurate when shooting any practical distance. As Grandfather walks by the room, he is only twenty yards away.
It seems that Tim can kill Grandfather. Every condition is optimal for a perfect shot that would kill him instantly. There is, however, the outstanding fact that Grandfather dies in his bed in 1957. It cannot be that both Tim murders Grandfather in 1921 and Grandfather dies of natural causes in 1957. Since we know Grandfather dies in his sleep in 1957, then it must be the case that Tim does not kill Grandfather in 1921. Now, it seems that Tim can't kill Grandfather.
This presents us with what looks like a contradiction. According to Arguments A and B listed above, Tim can kill Grandfather and Tim can't kill Grandfather. So which is right? Here are three ways out.
Tim Can't Kill Grandfather
One tempting option is to object to Argument A. Logic being what it is, the universe will not allow a contradiction to be created.
… [H]ow about the assassination of the grandparents? Could this extravagant crime be committed using the time machine? The answer is a categorical no (Novikov 1998, 263).
Grandfather didn’t die in 1921; he didn't die until 1957 and even then died of natural causes. So, if Tim were to try to kill Grandfather, he would fail. Tim might lose his nerve, or it is possible that the rifle will not go off when he fires it, or even after all of his practicing, Tim will fire a shot and miss. Tim cannot kill his grandfather. Time travelers “can’t change any past fact whatsoever” (Le Poidevin 2005, 179).
Tim Can Kill Grandfather
A second way out approach attempts to show that Argument B is unsound. The challenge is to the second premise of this argument: It doesn't follow from the fact that Tim doesn't kill Grandfather that Tim can't kill Grandfather. There are lots of things that we don't do but that we could have done. This approach is defended by Paul Horwich (1975).
A little elementary modal logic helps explain why the second premise of Argument B can seem true. Using a wide-scope reading, the second premise says that, necessarily, (if Tim didn't kill Grandfather, then Tim doesn't kill Grandfather). While this is true, when inserted back into the argument, the argument becomes invalid. Using a narrow-scope reading, the second premise says that if Tim didn't kill Grandfather, then, necessarily, (Tim doesn't kill Grandfather). This results in a valid argument, but, on this reading, the premise is false. According to this premise, Grandfather's actually dying of natural causes is enough to ensure that, necessarily, Tim does not kill him, which is not true. There are other possible worlds in which Grandfather didn't die in 1957, instead Tim kills him in 1920. Just because he didn't, doesn't mean he can't.
Context Dependence
Another answer to the paradox comes from David Lewis (1976). He states that the word 'can' is context dependent. This means that a sentence including the word 'can' may be true in some contexts and false in others. "To say that something can happen means that its happening is compossible with certain facts" (p. 150). He believes that 'x can do y' is true in context C if and only if x's doing y is consistent with certain facts picked out by C. Lewis does not elaborate about what he believes these "certain facts" are, but John Carroll (in press) suggests that these facts are the common ground of the context, which are the facts presumed to be shared information of the conversation, the presuppositions and suppositions.
Depending on the context, either Argument A or Argument B will be unsound. In conversations where we suppose that Grandfather died in 1957 of natural causes, Argument A will turn out unsound because 'Tim can't kill Grandfather' is true and therefore 'Tim can kill Grandfather' is false. In contexts where we don't suppose or presuppose any post-1920 information about Grandfather or Tim, then Argument B will be unsound because 'Tim can kill Grandfather' is true and 'Tim can't kill Grandfather' is false. What Tim is able to do relative to one set of facts he is not able to do relative to another. To get a contradiction, both arguments have to be sound in the same context. Since this is not the case, The Grandfather Paradox is not a significant challenge to the possibility of time travel.
Conclusions
Each of the three resolutions to The Grandfather Paradox seems plausible and yet very different from the other two. The second resolution finds fault with Argument B. This is very different from the first resolution, which finds fault with Argument A. Lewis shows why these two resolutions can seem appealing, and he also agrees that ultimately there is no contradiction. The only difference is that, with his approach, where the fault lies depends on the context of the argument.
References and Further Reading
Carroll, John. "Context, Conditionals, Fatalism, Freedom, and Time Travel." Topics in Contemporary Philosophy, Volume 6: Time and Identity. Eds. J. Keim-Campbell and M. O'Rourke. Cambridge: MIT Press, in press.
Horwich, Paul. "On Some Alleged Paradoxes of Time Travel." Journal of Philosophy 72 (1975): 432-444.
Le Poidevin, Robin. Travels in Four Dimensions. Oxford: Oxford University Press, 2005.
Lewis, David. "The Paradoxes of Time Travel." American Philosophical Quarterly 13 (1976): 145-152.
Novikov, Igor. The River of Time. Cambridge: Cambridge University Press, 1998.
Thomas, Stephen. "A Modal Muddle." Determinism, Free Will, and Moral Responsibility. Ed. G. Dworkin. Englewood Cliffs: Prentice Hall, 1970: 141-148.