Classic logic has nothing to do with time, let alone time travel. You might as well ask if the color yellow is consistent with time travel.
There's a branch of math known as "temporal logic", which allows one to make deductions that have to do with time, and the assumptions of temporal logic are that time behaves as we understand it, that is, without time travel.
All this means nothing about the possibility or impossibility of time travel, though. If time travel turns out to be possible (which I doubt), then we'll just have to come up with laws that take that into account.