Pointex

(This post is a continuation of the post Oscar on the island of two truths.)

It was a funny little dwarf called Pointex that explained to Oscar all the basics about the reasoning on the island.
"Are the queen and the king very smart?" asked Oscar.
"Of course they're smart! Some even believe that the queen and the king are right about everything!" said Pointex proudly.

Exercise 3. Show that if the queen and the king are right about everything then they must agree on everything.

Exercise 4. Note that if the queen and the king both believe p then they don't agree on p!

"Is there a way to tell who agrees with who?" asked Oscar. "It seems to me that the queen and the king don't agree on anything!" He was astonished by the strange relations that seemed to rule the island.
"Oh, no, of course they do!" said Pointex who was now amusing himself by jumping around Oscar balancing on his left foot.

Exercise 5. Show that if the queen and the king agree on a proposition p, then all natives that have an opinion on p agree on it.

Exercise 6. Show that if any native assigns value 0 to a proposition p, then all natives that have an opinion on p assign 0 to it.

"In fact," continued Pointex proudly, "some believe that our queen and king agree on everything!"
"But wouldn't that imply that the queen and the king are loyal to each other?" asked Oscar.

Exercise 7. Show that if the queen and the king agree on everything then they are loyal to each other. Moreover, if that is the case then all natives are loyal to both of them.

Exercise 8.  Show that if the queen is loyal to the king then the king is also loyal to the queen, and vice versa.

Exercise 9.  Show that if there exists a native that is loyal to the queen but not to the king, then the king is not loyal to the queen. The same assertion holds if there exists a native that is loyal to the king but not to the queen.

"Are you loyal to both the queen and the king?" asked Oscar.
Pointex looked at him with contempt. "What a ridiculous thing to ask! Certainly not."
The dwarf stopped jumping around Oscar and looked at the boy. Oscar looked at the little man. He was about 10 inches high, with messy brown hair and a long beard, wearing the most ridiculous clown cap that Oscar had ever seen.
Oscar was determined to get some more information. "Could you at least tell me whether the queen and the king know that they're loyal to themselves?" he asked.
"Well, of course they know that!" replied Pointex.
Oscar wanted to ask another question but when he turned around, Pointex was already gone.

Oscar met him again that afternoon. The funny dwarf was standing in front of the town fountain, amusing himself by throwing his cap high up into the air.
Oscar decided to clarify what was bothering him right away.
"You said that a proposition can be Q-TRUE, K-TRUE or FALSE," he said.
"Yes, of course. What else!" replied Pointex as he threw the cap high up.
"Well, you also said that each native was loyal either to the queen or to the king. I was just wondering ... Assume that A is loyal to the queen. Then what is the truth value of the statement: A is loyal to the queen?"
"Q, of course," answered Pointex as he threw the cap up again.
"And what if A is not loyal to the queen? What is then the truth value of the statement: A is loyal to the queen?"
He barely finished his question as something fell over his face and covered his eyes. It was the funny cap.
"Thanx," said Pointex as Oscar handed him the cap. "The value is 0, of course."
"Can the truth value of the statement: A is loyal to the queen. be K in any case?"
"Well, what do you think? Of course not! What a ridiculous thing to ask!" replied Pointex.

Exercise 10.  Assume that A is loyal to the queen and consider the following propositions:

a: A is loyal to the queen.
a': A is loyal to the king.

By what Pointex just told Oscar, the value of proposition a is Q and the value of proposition a' is 0. Show that if anybody on the island knows that A is not loyal to the king (ie. assigns value 0 to a'), then everybody that has an opinion on a' knows that.

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