Symbolic Logic

CHAPTER 8: STATEMENT LOGIC: PROOFS 109 8. 1. ~C → (N ∨ I) 2. C → M 3. ~M 4. T → ~I 5. T ∴ N 6. ~I 4, 5, MP 7. ~C 2, 3, MT 8. N ∨ I 1, 7, MP 9. N 6, 8, DS 9. 1. L ∨ F 2. L → C 3. C → T 4. F → P 5. P → S 6. I → ~T 7. I ∴ S 8. ~T 6, 7, MP 9. ~C 3, 8, MT 10. ~L 2, 9, MT 11. F 1, 10, DS 12. P 4, 11, MP 13. S 5, 12, MP 10. 1. M ∨ ~M 2. M → R 3. ~M → K 4. (R ∨ K) → E 5. E → D ∴ D 6. R ∨ K 1, 2, 3, CD 7. E 6, 4, MP 8. D 6, 7, MP Exercise 8.2 Part A: Annotating *1. 1. ~~A → B ∴ A → B 2. A → B 1, DN 2. 1. ~C → ~D ∴ D → C 2. D → C 1, Cont 3. 1. ~(E • ~D) ∴ D ∨ ~E 2. ~E ∨ ~~D 1, DeM 3. ~E ∨ D 2, DN 4. D ∨ ~E 3, Com *4. 1. ~(E ∨ D) ∴ ~D 2. ~E • ~D 1, DeM 3. ~D 2, Simp 110 SOLUTIONS MANUAL 5. 1. ~A • [(A ∨ B) ∨ C] ∴ B ∨ C 2. ~A 1, Simp 3. (A ∨ B) ∨ C 2, Simp 4. A ∨ (B ∨ C) 3, As 5. B ∨ C 4, 2, DS 6. 1. F • (G • R) ∴ G • F 2. (F • G) • R 1, As 3. F • G 2, Simp 4. G • F 3, Com *7. 1. [(P → Q) → R] • (~Q → ~P) ∴ ~~R 2. ~Q → ~P 1, Simp 3. P → Q 2, Cont 4. (P → Q) → R 1, Simp 5. R 4, 3, MP 6. ~~R 5, DN 8. 1. [~(S • T) ∨ ~~U] • (T • S) ∴ U 2. T • S 1, Simp 3. S • T 2, Com 4. ~(S • T) ∨ ~~U 1, Simp 5. ~~(S • T) 3, DN 6. ~~U 4, 5, DS 7. U 6, DN 9. 1. ~W ∨ (~X ∨ ~Y) ∴ ~Y ∨ ~(W • X) 2. (~W ∨ ~X) ∨ ~Y 1, As 3. ~(W • X) ∨ ~Y 2, DeM 4. ~Y ∨ ~(W • X) 3, Com *10. 1. [~O → (~M → ~N)] • ~(N → M) ∴ O 2. ~O → (~M → ~N) 1, Simp 3. ~(N → M) 1, Simp 4. ~(~M → ~N) 3, Cont 5. ~~O 2, 4, MT 6. O 5, DN 11. 1. ~~P • ~P ∴ W 2. ~~P 1, Simp 3. ~~P ∨ W 2, Add 4. P ∨ W 3, DN 5. ~P 1, Simp 6. W 4, 5, DS 12. 1. ~T ∴ ~(T • ~~S) 2. ~T ∨ ~S 1, Add 3. ~(T • S) 2, DeM 4. ~(T • ~~S) 3, DN *13. 1. ~A ∴ ~[(B • C) • A] 2. ~(B • C) ∨ ~A 1, Add 3. ~[(B • C) • A] 2, DeM CHAPTER 8: STATEMENT LOGIC: PROOFS 111 14. 1. (S → G) • (G → T) ∴ ~T → ~S 2. S → G 1, Simp 3. G → T 1, Simp 4. S → T 2, 3, HS 5. ~T → ~S 4, Cont 15. 1. ~R → ~S ∴ ~P ∨ (S → ~~R) 2. S → R 1, Cont 3. S → ~~R 2, DN 4. ~P ∨ (S → ~~R) 3, Add Part B: Correct or Incorrect? *1. DeM 2. Cont 3. Incorrect. If we apply DeM to ~(F ∨ G), we get ~F • ~G. *4. DeM 5. Com 6. Cont *7. Incorrect use of DeM. Correct sequence: from ~S ∨ T to ~S ∨ ~~T by DN; from ~S ∨ ~~T to ~(S • ~T) by DeM. 8. Cont 9. Incorrect use of Cont; requires a step of DN as well as contraposition. *10. Cont 11. Incorrect; requires a step of association followed by a step of commutation. 12. DeM *13. Com 14. DeM 15. DeM *16. As 17. DeM 18. DN *19. DeM 20. Incorrect. The given conclusion follows validly, but only by a sequence of steps. Part C: Proofs *1. 1. ~(C • D) 2. ~ C → S 3. ~D → T ∴ S ∨ T 4. ~C ∨ ~D 1, DeM 5. S ∨ T 2, 3, 4, CD 2. 1. (W → U) • ~X ∴ ~U → ~W 2. W → U 1, Simp 3. ~U → ~W 2, Cont 3. 1. F → ~G 2. G ∴ ~F 3. ~~G 2, DN 4. ~F 1, 3, MT *4. 1. ~(~A ∨ B) ∴ A 2. ~~A • ~B 1, DeM 3. A • ~B 2, DN 4. A 3, Simp 112 SOLUTIONS MANUAL 5. 1. (~P → Q) • ~Q ∴ P 2. ~P → Q 1, Simp 3. ~Q 1, Simp 4. ~~P 2, 3, MT 5. P 4, DN 6. 1. ~(N ∨ M) 2. ~L → (M ∨ N) ∴ L 3. ~(M ∨ N) 1, Com 4. ~~L 2, 3, MT 5. L 4, DN *7. 1. (A ∨ B) ∨ C 2. ~A ∴ C ∨ B 3. A ∨ (B ∨ C) 1, As 4. B ∨ C 2, 3, DS 5. C ∨ B 4, Com 8. 1. (W • ~X) ∨ (Y • Z) 2. (~X • W) → U 3. (Y • Z) → T ∴ U ∨ T 4. (~X • W) ∨ (Y • Z) 1, Com 5. U ∨ T 4, 2, 3, CD 9. 1. ~(S ∨ R) 2. P → R ∴ ~P 3. ~S • ~ R 1, DeM 4. ~R 3, Simp 5. ~P 2, 4, MT *10. 1. F → (G • H) 2. (H • G) → J ∴ F → J 3. (G • H) → J 2, Com 4. F → J 1, 3, HS 11. 1. K ∨ (L ∨ S) 2. ~(K ∨ L) ∴ S 3. (K ∨ L) ∨ S 1, As 4. S 2, 3, DS 12. 1. ~P 2. ~(P ∨ Q) → ~R 3. ~Q ∴ ~R 4. ~P • ~Q 1, 3, Conj 5. ~(P ∨ Q) 4, DeM 6. ~R 2, 5, MP *13. 1. ~S → (T • U) 2. (~S → X) → ~Z 3. (U • T) → X ∴ ~Z 4. (T • U) → X 3, Com 5. ~S → X 1, 4, HS 6. ~Z 2, 5, MP

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