The Self-Visitation Paradox
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Consider Ted. Suppose that, on Monday at noon, Ted was sitting. The next Friday, Ted travels back to Monday at noon and stands while his younger self is sitting. Can Ted be both sitting and standing? Time traveling seems to have provided us a way in which he can, though it seems obvious that a person cannot be both sitting and standing. Below are a few ways that philosophers have found to resolve this paradox.
No-Contradiction
The no-contradiction solution accepts that the sitting and the standing can occur simultaneously, done by a single person. On this view, the apparent paradox stems from the natural assumption that, if Ted is sitting, then it is not the case that Ted is standing. But, maybe that assumption is natural for us only because we are used to thinking about situations that do not involve time travel. The no-contradiction solution asserts that sitting and standing are not mutually exclusive actions. While there would be a contradiction if Ted were both sitting and it were not the case that he was sitting, or standing and not the case that he was standing, it is not contradictory that Ted is both sitting and standing. So, the self-visitation scenario may not be the problem that it first may have seemed.
Relativizer
Relativists believe that location is the key to finding consistency. Perhaps the statements 'Ted is sitting' and 'Ted is standing' have implicit locations attached to them. So, the full meaning of these statements might be given by 'Ted is sitting over here' and 'Ted is standing over there'. Just as Ted can be sitting at one time and standing at another time, so can he be sitting in one place and standing in a different place. So, Ted's time travel scenario is not paradoxical.
The relativizer approach, however, takes for granted that an object can exist in two separate locations simultaneously. Ted is supposed to simultaneously be here and over there. This is as puzzling as his being both sitting and standing; it is another paradoxical aspect of the situation. But with this aspect of the situation relativization to location won't help. Perhaps the relativizer must accept the no-contradiction approach to the self-visitation paradox, at least with regard to the property of location.
Spatial Parts
Maybe the trouble arises from the thought that the sitting and the standing Teds are one person. Though they both answer to 'Ted', one might maintain that, strictly speaking, they are only two parts of Ted. Consider Younger Ted and Older Ted, and don't assume they are one person. Maybe Younger Ted is one spatial part and Older Ted is another spatial part of Ted. Different spatial parts of an object may have different properties without contradiction. Just as Ted's head might be bald and his chest hairy, a spatial part of Ted (Younger Ted) may be sitting while another spatial part (Older Ted) is standing. The surprising and sudden appearance of a twice as heavy, four-eyed, two-nosed, spread out Ted might be considered an unexpected effect of time travel, not something that is inconsistent. (This view is offered by Ned Markosian (2004) as a parallel alternative to the perdurantist views discussed below.)
Perdurantism
So far, we have been working with a natural assumption about how objects persist through time known as endurantism. In other words, we have thought of Ted as an object that exists wholly at any particular instant. Ted endures through time by continuing to exist, just growing and changing as time passes. Endurantism, however, is not the only perspective, and the solution to the self-visitation paradox may lie in denying it. The alternative is known as perdurantism.
Instead of Ted, let's consider Elvis Presley for a moment. In 1958, as a rising star, he is good-looking and in shape. Later, by 1975, he is a very overweight man. One might argue that since Younger Elvis is slender and Older Elvis is rotund, they cannot be identical--that they are not one person. The perdurantist would say that the part of Elvis that exists earlier is thinner than the part that exists later, and that they are just temporal parts of Elvis (who is the collection of all his temporal parts). Endurantists hold instead that, while Elvis wholly exists at each of those times, and continues to exist from one time to the next, his properties change.
Since perdurantism allows for contradicting properties in a non-time-traveling situation, perdurantism appears to be an easy way out of our self-visitation problem. Maybe, on Monday at noon, there is a temporal part of Ted, a part that happens to be sitting; maybe there is another temporal part of Ted then too, a standing part.
Sounds good, but that's too quick. In the self-visitation case, we cannot be concerned with two different temporal parts as we were with the case of Elvis. The pertinent parts of Ted are simultaneous; they both exist on Monday at noon. It looks like the perdurantist must rely on aspects of the relativizer, the no-contradiction, or the spatial-parts approach: A perdurantist could hold that the sitting part and the standing part are two distinct spatial parts of one temporal part of Ted — that's the usual perdurantist way out (e.g., Lewis 1976, 147; Sider 2000, 101). Alternatively, one could hold that this temporal part of Ted is sitting only relative to one location and standing only relative to another. For a third option, a perdurantist could hold that it is not contradictory for a temporal part to be both sitting and standing.
References and Further Reading
Haslanger, Sally. "Persistence Through Time." Oxford Handbook of Metaphysics. Eds. M. Loux and D. Zimmerman. Oxford: Oxford University Press, 2003.
Lewis, David. "The Paradoxes of Time Travel." American Philosophical Quarterly 13 (1976): 145-152.
Markosian, Ned. "Two Arguments From Sider's Four-Dimensionalism." Philosophy and Phenomenological Research 68 (2004): 665-673.
Sider, Theodore. Four Dimensionalism. Oxford: Clarendon Press, 2000.
Simon, Jonathan. "Is Time Travel a Problem for the Three-Dimensionalist?" Monist 88 (2005): 353-361.