# Chapter 7

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## While you read Edit

### Questions Edit

- Why does DRH use fragments of Zen koans as propositions?
- 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>**? - 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*. - 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?
- Consider the four theorems on p. 197. State them as "koans". Then derive at least one of them using the Propositional Calculus.
- Is the Propositional Calculus consistent? Is it complete?

### Characters for copying and pasting Edit

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:

⟨ ⟩ [ ] ∧ ∨ ⊃ ~