Chapter 3
- Provide an exceedingly clear presentation of the expressive completeness of a propositional language containing only the Scheffer stroke (or alternatively Peirce's arrow). The proof should take a route different from the one in the book (or the one taken in class).
- Is there an English operators other than 'not both' or 'neither nor' that is expressively complete with respect to propositional logic? Prove it.
Chapter 4
- Prove exercise 4.4.2, #9.
Chapter 5 (extra metatheorems)
[note: the letters below should be read as our propositional metavariables]
- A conditional with an inconsistent antecedent is valid.
- if p and q are equivalent formulas, then ~p and ~q are equivalent formulas as well.
- If a biconditional p <-> q is valid, then the set {p, ~q} is inconsistent.
- If a set of formulas is inconsistent, then those formulas jointly entail any formula.
- Prove that p --> q and ~p v q are logically equivalent.
- By mathematical induction, prove that any inconsistent formula contains at least one negation.
- by mathematical induction, prove that every formula has an even number of parentheses.
- Prove that if p1, ..., pn |- q is a valid sequent, then (p1 & ... & pn) --> q is a valid formula.