Chomsky has a cat called Mimi who sleeps at the bottom of his bed at night.

When Chomsky is asleep everything in his dreams that he believes to be true is in fact false.  In dreams he believes that chairs can talk and that books can swim.

When he’s awake he knows that chairs can’t talk and that books can’t swim because eveything Chomsky believes when he’s awake is true

One night as the clock strikes midnight, Chomsky believes that both he and Mimi are asleep. Was Mimi in fact awake or asleep?

Answer.

Since all Chomsky waking beliefs are true, knowing that he was awake, he couldn’t have believed that both he and Mimi were asleep. Therefore he was asleep. So this means that his belief was false and he and Mimi were not both asleep. Since Chomsky was asleep, Mimi must have been awake. QED