Which Alice?

13
The Red King's Theory
"Right about what?" asked the King. "I never really had this
conversation with Humpty Dumpty; I only dreamed that I did!"
"I didn't mean the real Humpty Dumpty," replied Alice. "I meant
the Humpty Dumpty you dreamed about. It was he who was right!"
"Now, just a minute!" said the King, "what are you trying to tell
me—that there are two Humpty Dumptys, one the real one and the
other the one I dreamed about?"
Alice did not quite know what to say to this.
"Anyway," said the King, "in the meantime, 1 have thought of a
much better argument proving that I am awake—this argument
couldn't possibly be wrong; it must be right!"
"Now, that I should like to hear!" said Alice.
"Well," said the King, "to begin with, I hold the theory that
everyone in the world is of one of two types: Type A or Type B.
Those of Type A are totally accurate in their beliefs while they are
awake, but totally inaccurate while they are asleep. Everything they
believe while they are awake is true, but everything they believe
while they are asleep is false. People of Type B are the reverse:
Everything they believe while they are asleep is true, and everything
they believe while awake is false."
"What an extraordinary theory!" said Alice. "But what proof do
you have that it is correct?"
"Oh, I will later prove to you beyond any reasonable doubt that it
is correct, but for the time being, I want you to realize some of the
consequences of the theory. To begin with, the following two
propositions follow immediately:
"Proposition One—If, at a given time, a person believes he is awake,
then he must be of Type A.
"Proposition Two—If, at a given time, a person believes he is of
Type A, then he must be awake at that time."
The King then proved these two propositions to Alice's satisfaction;
at least Alice could find no flaw in the arguments.
A QUESTION Do Propositions 1 and 2 really follow from the
Red King's theory?