This is a series on the book Gödel, Escher, Bach: An eternal golden braid by Douglas Hofstadter.
Earlier diaries are here
Today we'll discuss Chromatic Fantasy and Feud (pages 177-180) and Chapter 7: The propositional calculus (pages 181-197).
This series usually goes up on Sunday morning, but I was on vacation last week an didn't write the diary.
From the overview:
Chromatic fantasy, and feud A short dialogue bearing hardly any resemblance, except in title, to Bach's Chromatic Fantasy and Fugue . It concerns the proper way to manipulate sentences to preserve truth - and in particular the question of whether there exist rules for usage of the word "and". This dialogue has much in common with the dialogue by Lewis Carroll.
Chapter VII: The Propositional Calculus It is suggested how words such as "and" can be governed by formal rules. Once again, the ideas of isomorphism and automatic acquisition of meaning by symbols in such a system are brought up. All the examples in this Chapter, incidentally, are "Zentences" - sentences taken from Zen koans. This is deliberately done, somewhat tongue in cheek, since Zen koans are deliberately illogical stories.
The dialogue, I think, stands on its own. But how about coming up with two sentences, each acceptable, that are nonsense when joined with an "and"?
I have relatively little to say about the chapter, as well. It's a fairly short and fairly dry chapter, which will be of a lot of use in later chapters. Make sure you understand it, or you will be lost later.
Page 181 - a puzzle? In GEB? Shocking!
Page 182 - spoiler .... don't read until you try on your own.
1 and 2 have angle brackets where they should not
3 needs another set of angle brackets somewhere
5 can't have the ^ after the ~
8 needs more brackets
Page 184 - this starts to get at the difference between TRUE and VALID. The PC generates atatementst that are TRUE, but they are not atomic. The PC says nothing whatever about the premises .... that is, the atoms. The premises can even (as shown later in the chapter) contain contradictions.
Page 185 - note the subtle but important difference between the final statement here and the one on page 184.
Page 192 - can we prove logic? Can we doubt it? Can you envision a universe where it doesn't apply? How does this relate to quantum mechanics?