Post #8: The end is nigh

Assignment 3 has just been completed (well, mostly). At this point, we just have to review everything and make sure that we're not leaving out comments or steps. I am putting off this task until later because it requires a lot of to-and-fro scanning, which isn't difficult - it's just tedious. Meanwhile, I have been working really hard this week to try to catch up with everything that has been going on. As the end of the term (and school year) comes to a close, everything is coalescing at the last moment and making the last few days EXTREMELY hectic.

I have started on my problem solving question, and wrote down some preliminary observations and notes. I will continue working on the problem later this week. Excuse any typos I may have made; I rushed the typing of the problem (which is quite long, as you can imagine, since it is part of a storybook). 

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Context - Problem Solving
After several hours of questioning, Alice gathered an enormous amount of data, which she recorded in her memorandum book. She took it all to Humpty Dumpty to see if he could explain it.

“It figures,” said Humpty Dumpty, looking through Alice’s notes, “it figures!”

“What do you mean by that?” asked Alice. “Is this White Knight untruthful?”

“The White Knights never lie, “ replied Humpty Dumpty.

“Then I don’t understand,” replied Alice, “I really don’t understand!”

“Of course not,” responded Humpty Dumpty contemptuously, “you don’t understand Looking-Glass logic!”

“And what is Looking-Glass logic?”

“The kind of logic used by Looking-Glass logicians,” he replied.

“And what is a Looking-Glass logician?” asked Alice.

“Why, one who uses Looking-Glass logic,” he replied. “Surely, you could have guessed that!”

Alice thought this over. Somehow, she didn’t find this explanation very helpful.

“You see,” he continued, “there are certain people here called Looking-Glass logicians. Their statements seem a bit bizarre until you understand the key -- which is really quite simple. Once the key is understood, the whole business makes perfect sense.”

“And what is the key?” asked Alice, more curious than ever.

“Oh, it would never do to tell you the key! However, I will give you some clues. In fact, I will give you the five basic conditions about Looking-Glass logicians from which you can deduce the key.  Here are the conditions:

1. A Looking-Glass logician is completely honest. He will claim those and only those statements which he actually believes.

2. Whenever a Looking-Glass logician claims a statement to be true, then he also claims that he does not believe the statement.

“Just a minute,” interrupted Alice. “Are you not contradicting yourself> According to the first condition, a Looking-Glass logician is always truthful. If, then, he claims a statement to be true, he must really believe that it is true. How then, without lying, can he claim that he doesn’t believe the statement?”
“Good question,” replied Humpty Dumpty. “However, I never said that a Looking-Glass logician is always accurate! Just because he believes something doesn’t mean that he necessarily knows that he believes it, nor even that he necessarily believes that he believes it. Indeed, it could happen that he erroneously believes that he doesn’t believe it.”

“You mean,” replied Alice, utterly astonished, “that a person can actually believe something, and yet believe that he doesn’t believe it?”

“With Looking-Glass lgoicains, yes,” replied Humpty Dumpty, ‘in fact with Looking-Glass logicians this always happens -- this is a direct consequence of the first two conditions.”

“How is that?” asked Alice.

“Well,” replied Humpty Dumpty, “Suppose he believes a statement to be true. Then, by Condition One, he claims the statement is true. Then, by Condition Two, he claims that he doesn’t believe the statement. Hence, again by Condition One, he must believe that he doesn’t believe the statement.

“Anyhow,” continued Humpty Dumpty, “I’m giving you too many hints! Let me finish my list of conditions, and then you should deduce the key to the entire mystery.”

3. Given any true statement, the Looking-Glass logician always claims that he believes the statement.

4. If a Looking-Glass logician believes something, then he cannot also believe its opposite.

5. Given any statement, a Looking-Glass logician either believes the statement or he believes its opposite.

“And that,” concluded Humpty Dumpty proudly, “is the entire list of conditions. From these you should be able to infer just which statements a Looking-Glass logician believes to be true and just which ones he believes to be false. I will now ask you some questions to test your understanding.”

QUESTION

Suppose he believes that all gryphons have wings. Does it follow that there are any gryphons?

We have the conditions: 

1. A Looking-Glass logician is completely honest. He will claim those and only those statements which he actually believes.

2. Whenever a Looking-Glass logician claims a statement to be true, then he also claims that he doesn’t believe the statement.

3. Given any true statement, the Looking-Glass logician always claims that he believes the statement.

4. If a Looking-Glass logician believes something, then he cannot also believe its opposite.

5. Given any statement, a Looking-Glass logician either believes the statement or he believes its opposite.

To do: tweed out the differences between claim and believe (because they are definitely different -- one is more declarative while the other isn't)

To do: translate things into symbols.

BELIEVE [TRUE FALSE] and CLAIM [TRUE FALSE] into predicates, in order to evaluate the statements above. 

I will follow up on this soon!