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Theories of time travel are riddled with questions about causality and paradoxes. Compared to other fundamental concepts in modern physics, time still not understood very well. Philosophers have been theorizing about the nature of time since era of the ancient Greek philosophers and earlier (See the main article on Philosophy of space and time). Some philosophers and physicists who study the nature of time also study the possibility of time travel and its logical implications. The probability of paradoxs and their possible solutions are often considered.

For more information on the philosophical considerations of time travel, consult the work of David Lewis or Ted Sider. For more information on physics-related theories of time travel, consider the work of Kurt Gödel (especially his theorized universe) and Lawrence Sklar.

The Grandfather Paradox
One subject often brought up in philosophical discussion of time is the idea that, if one were to go back in time, paradoxes could ensue if the time traveler were to change things. The best examples of this are the grandfather paradox and the idea of autoinfanticide. The grandfather paradox is a hypothetical situation in which a time traveler goes back in time and attempts to kill his grandfather at a time before his grandfather met his grandmother. If he did so, then his father never would have been born, and neither would the time traveler himself, in which case the time traveler never would have gone back in time to kill his grandfather, and so on.

Autoinfanticide works the same way, where a traveler goes back and attempts to kill himself as an infant. If he were to do so, he never would have grown up to go back in time to kill himself as an infant, and so on.

This discussion is important to the philosophy of time travel because philosophers question whether these paradoxes make time travel impossible. Some philosophers answer the paradoxes by arguing that if someone were to go back in time and attempt kill his infant self, a series of coincidences would ensue which would prevent the murder, and that only with these coincidences can time travel be possible.

Theory of Compossibility
David Lewis’ analysis of compossibility and the implications of changing the past is meant to account for the possibilities of time travel in a one-dimensional conception of time without creating logical paradoxes. Consider Lewis’ example of Tim. Tim hates his grandfather and would like nothing more than to kill him. The only problem for Tim is that his grandfather died years ago. Tim wants so badly to kill his grandfather himself that he constructs a time machine to travel back to 1955 when his grandfather was young and kill him then. Assuming that Tim can travel to a time when his grandfather is still alive, the question must then be raised; Can Tim kill his grandfather?

For Lewis, the answer lies within the context of the usage of the word "can". Lewis explains that the word "can" must be viewed against the context of pertinent facts relating to the situation. Suppose that Tim has a rifle, years of rifle training, a straight shot on a clear day and no outside force to restrain Tim’s trigger finger. Can Tim shoot his grandfather? Considering all of the facts that I have just listed, it would appear that Tim can in fact kill his grandfather. In other words, all of the contextual facts are compossible with Tim killing his grandfather. However, when reflecting on the compossibility of a given situation, we must gather the most inclusive set of facts that we are able to.

Consider now the fact that Tim’s grandfather died in 1993 and not in 1955. This new fact about Tim’s situation reveals that him killing his grandfather is not compossible with the current set of facts. Tim cannot kill his grandfather because his grandfather died in 1993 and not when he was young. Thus, Lewis concludes, the statements "Tim doesn’t but can, because he has what it takes," and, "Tim doesn’t, and can’t, because it is logically impossible to change the past," are not contradictions, they are both true given the relevant set of facts. The usage of the word "can" is equivocal: he "can" and "can not" under different relevant facts. So what must happen to Tim as he takes aim? Lewis believes that his gun will jam, a bird will fly in the way, or Tim simply slips on a banana peel. Either way, there will be some logical force of the universe that will prevent Tim every time from killing his grandfather.

Problems with this logic
This logic is not without its own set of problems. Lewis' conception of compossibility implies consequences that relate outside of just the grandfather paradox. Lewis demands that we examine all of the pertinent facts regarding the situation in order to delineate the compossibility of a given event. But, when Lewis claims that Tim cannot kill his grandfather because of the fact that his grandfather died in 1993, this sort of counterfactual consideration opens the way for difficulties with Tim’s time travel to begin with. No matter how sly or subtle Tim acts when he has traveled to the past, he has been setting up counterfactuals since the instance he arrived.

In other words, if Tim cannot kill his grandfather in 1955 due to the fact that his grandfather died much later in 1993 then we are allowed to stop Tim from doing a number of things in the past. For example, Tim could not have bought a rifle in the past because any rifle that he would have chosen could be coupled with the fact that that particular rifle was bought by Mr. Smith in 1955 and not Tim. This sort of objection may seem at first arbitrary, though we must consider it as rather probable. Thus, Tim buying his gun to shoot his grandfather is not compossible with the most inclusive set of facts regarding the gun he has chosen. One can follow this line of thinking to a complete dismissal of the possibility of Tim traveling back in time at all. No matter how unobtrusive Tim is in the past, the very fact of his existence in the past has countered many of the facts about the past - the most glaring being the fact that Tim was not in existence in 1955. Thus, from this perspective we can find a very large gap in Lewis’ theory which has yet to be addressed by Lewis.