56 Logic as a Tool References for further reading For further discussion and examples on derivations in axiomatic systems for propositional logic, see Tarski (1965), Shoenfield (1967), Hamilton (1988), Fitting (1996), and Mendelson (1997) Exercises 2.2.1 Show that the result of uniform substitution of a propositional formula for a propositional variable in a propositional formula is again a propositional formula (Hint: use induction on the formula in which the substitution is performed.) 2.2.2 Derive the following in H without using the Deduction Theorem (NB: some of these are simple, others are quite tricky The purpose of this exercise is to make you appreciate the Deduction Theorem.) (a) p H p ∨ q ; q H p ∨ q (f) H (p → q ) → ((q → r) (b) p, q H p ∧ q → (p → r)) (c) H (p ∧ q ) → (q ∧ p) (g) H (¬p → ¬q ) → (q → p) (d) H (p ∨ q ) → (q ∨ p) (h) H (q → p) → (¬p → ¬q ) (e) H p → p 2.2.3 Prove the principle of induction on derivations in H by using induction on lengths of derivations 2.2.4 Complete the proof of the Deduction Theorem by using the principle of induction on derivations in H 2.2.5 Prove that if Γ, A H B and Γ, B H C then Γ, A (Hint: use the Deduction Theorem.) 2.2.6 If A H B then B → C H H C A → C 2.2.7 Prove, by induction on derivations in H, that every theorem of H is a tautology 2.2.8 More generally, using induction on derivations in H, prove that H is sound, that is, for every set of formulae Γ and a formula A, if Γ |= A then Γ H A 2.2.9 Derive the following in the axiomatic system H using the Deduction Theorem, where P, Q, R are any formulae (Hint: for some of these exercises you may use the previous exercises.) (a) (b) (c) (d) (e) (f) (g) (h) P H P ∨ Q; Q H P ∨ Q P, Q H P ∧ Q H (P ∧ Q) → (Q ∧ P ) H (P ∨ Q) → (Q ∨ P ) P → (Q → R) H Q → (P → R) (P ∧ Q) → R H P → (Q → R) P → (Q → R) H (P ∧ Q) → R If ¬P H ¬Q then Q H P (i) (¬P → ¬Q) H (Q → P ) (j) If ¬P, Q H ¬Q then Q H P (k) If Q H P then ¬P H ¬Q (l) (Q → P ) H (¬P → ¬Q) (m) ¬¬P H P (n) P H ¬¬P (o) H P ∨ ¬P (p) H ¬(P ∧ ¬P )