Consider the Truth-Teller Paradox:
- This sentence is true.
- This sentence is false.
This kind of twist can be found in other cases. One might be vaguely worried about the set of all sets that contain themselves, and then make the twist and get the clearly paradoxical set of all sets that do not contain themselves. Or one might be worried about the causal loops that time travel would permit, say one's getting the plans for the time machine from one's future self, and then after building the time machine going back in time to hand those plans to one's then-past self. There is something fishy about such a causal loop. So you give it a twist, and you turn it into a clearly paradoxical story about shooting your grandfather before his children are conceived. I think one can argue that in this same way, Thomson's Lamp Paradox is a twist on Zeno's Achilles Paradox, and the Grim Reaper Paradox is a twist on Zeno's Dichotomy.
I wonder if there is anything interesting and general one can say about the logical structure of the twist. (There may be something in the literature.) In particular, I am curious whether one can infer the impossibility of the untwisted situation from the impossibility of the twisted situation.