Dr Effingham argues that time travel is logically possible—that is, it can’t be ruled out on the grounds of any contradiction.
The best theory allowing for time travel is one whereby you are unable to change the past. For instance, if you go back to kill your grandfather, you will fail. Something will stop you. You’ll have a heart attack. Or miss. Or just shoot the wrong person!
Whilst this theory has been defended before, Dr Effingham discusses how this feeds into issues concerning probability and decision making. In the case of you killing your grandfather, since you must fail, the chances of the events that could stop you all go up. For instance, your chance of having a heart attack suddenly sky rockets! Moreover, it doesn’t matter whether the change you’re bringing about is huge (e.g. shooting your grandfather) or small (e.g. shifting a grain of sand or an air molecule so that, where it was once in one place, it is now in a different place).
All such things are as contradictory as one another; all such things are impossible; all such things increase the chances of some awful accident befalling you. The upshot is that anyone even trying to travel in time will come undone. Something will stop them from using their time machine, probably killing them if they’re really persistent.
And this applies even to unintentional time travel. Some people have speculated that events of such unintentional time travel might come about in particle accelerators like CERN (and that, without realising it, a particle might skip back in time a few nanoseconds). Even that sort of time travel must be prevented. Dr Effingham’s ultimate conclusion is that, because of all of this, we should police those high energy experiments to ensure that no time travel event takes place, lest we risk some massive accident (or worse) befalling us.
If you would like to know more about Dr Effingham’s research, he has a TedX talk on just this topic.