In other words it is against the view according to which one's identity can be divided into fragments. The focus is on the unity of the being. This means that today I am the person that I was yesterday and the person I will be tomorrow even if my characteristics change as I move on the time axis.
The main difference resides in the very approach to the matter, for one depends on Time while the other judges things from an atemporal point-of-view.
If we are to accept the perdurantist conception, one could probably better understand how it is possible for one to be divided on the time axis. From this point Lewis would be an endurantist as opposed to Horwich who would be a perdurantist. However, since both of them seem to have a common goal under these circumstances, and through this I refer to the fact they both prove the grandfather paradox arguments to be false, it might be argued that their difference of conceiving the entire situation is irrelevant.
The argument that Lewis brings when he discusses the importance of context is supported by the very conception that people are entities which basically remain the same in time. The power and the potential of their actions changes as the circumstances which they find themselves in change.
Therefore, the changes occur outside the being and not inside it. This would explain why the adepts of this particular philosophical view put so much emphasis on the "constancy" (and unity) of a person's identity throughout time.
It might be argued that under these circumstances the potential of the human being's actions is always fulfilled (or not) due to the circumstances the being finds herself in. this means that if Tim goes back in time with a will to kill his grandfather and he is not only...
However, it is absurd to say that our deeds do not have consequences. Therefore, if Tim kills his grandfather before the birth of one of his own parents, then the direct consequence is that Tim also ceases to exist. But it must be mentioned that he ceases to exist even before he can take into consideration the desire to kill his grandfather. It seems that the argument is cyclical and self contradictory.
The only alternative of interpretation that we have is that while circumstances may play a very important role upon the realization of our potential and of our desires, so does our will. This is how the second argument can be explained. Tim may go back in time with a desire to kill the grandfather, he may have the necessary means and the opportunity to do it. If he does not do it, it does not necessarily mean that the circumstances suddenly became unfavourable. It may simply be that he changed his mind about the whole thing.
A person changes his desires and thoughts according to factors which nobody can entirely control and yet maintain the basic identity constant. This does not break the person into a multitude of beings, just like it does not state that one remains forever the same. On the contrary it demonstrates that a person is something beyond the sum of selves on the time axis, just like she or he is himself/herself despite the changes he/she undergoes.
Bibliography
Temporal parts." Stanford Encyclopedia of philosophy. October 1, 2004. 11 december 2008 http://plato.stanford.edu/entries/temporal-parts/
The time travel paradox. A time travel website. 11 December 2008. http://www4.ncsu.edu/~carroll/time_travel/grandfather.html
