S/P/20. Does David Lewis succeed in showing that time-travel is logically possible?

Einsteinian physics has shown time-travel to be theoretically plausible. All it would take is for an object to be spun faster than the speed of light, and relativity would send it spiralling towards an earlier time-frame. David Lewis attempts to eliminate any potential contradictions arising from time-travel in "Paradoxes of Time-Travel". How successfully does Lewis demonstrate the logical possibility of time-travel? In order to answer this question, we will firstly examine the 'Grandfather paradox' objection to time-travel and Lewis' solution to this before detailing Horwich's probability argument, then finally Smith's attempt to prove the possibility of time-travel.