Chapter 7

While you read

Questions

1. Why does DRH use fragments of Zen koans as propositions?
2. Consider the mini-dialogue on p. 191-192. Prudence wants to be convinced that P and ~P cannot both be theorems. Would Prudence be satisfied with a proof of the proposition ~<P ^ ~P>?
3. The "Fantasy Rule" is written informally in English. Re-state it as a pattern-matching rule, somewhat like the rules of the MIU-system and the pq-system. You should be able to state it as a meta-rule of the form "If X is a sequence of lines that follow these rules, then Y is a theorem", by filling in X and Y.
4. Why does the "second De Morgan's Rule" (p. 193) have to remain outside the system? Why can't it be proven as a theorem?
5. Consider the four theorems on p. 197. State them as "koans". Then derive at least one of them using the Propositional Calculus.
6. Is the Propositional Calculus consistent? Is it complete?

Characters for copying and pasting

Some of the characters DRH chose to use are inconvenient to type. In a pinch, you can copy and paste them. Here are all the symbols of the Propositional Calculus, besides the letters for propositions:

   ⟨ ⟩ [ ] ∧ ∨ ⊃ ~