Symbolic Logic

CHAPTER 8: STATEMENT LOGIC: PROOFS 105 *13. 1. (Z • A) ∨ ~Y 2. (Z • A) → U 3. W ∨ ~U 4. ~W ∴ ~Y 5. ~U 3, 4, DS 6. ~(Z • A) 2, 5, MT 7. ~Y 1, 6, DS 14. 1. (D • E) ∨ F 2. F → C 3. (D • E) → ~B 4. (~B ∨ C) → (A → P) 5. ~P ∴ ~A 6. ~B ∨ C 1, 3, 2, CD 7. A → P 4, 6, MP 8. ~A 7, 5, MT 15. 1. O → N 2. ~M 3. S → R 4. P → O 5. R → P 6. (S → N) → (M ∨ L) ∴ ~P ∨ L 7. S → P 3, 5, HS 8. S → O 7, 4, HS 9. S → N 8, 1, HS 10. M ∨ L 6, 9, MP 11. L 10, 2, DS 12. ~P ∨ L 11, Add *16. 1. (~M ∨ L ) → (~A → B) 2. ~S → T 3. R → ~S 4. ~M • J 5. R ∨ ~A ∴ T ∨ B 6. R → T 3, 2, HS 7. ~M 4, Simp 8. ~M ∨ L 7, Add 9. ~A → B 1, 8, MP 10. T ∨ B 5, 6, 9, CD 17. 1. F → ~G 2. ~~G • O 3. (~F • ~~G) → [(~H → E) • (C → F)] 4. C ∨ ~H ∴ E 5. ~~G 2, Simp 6. ~F 1, 5, MT 7. ~F • ~~G 5, 6, Conj 8. (~H → E) • (C → F) 3, 7, MP 9. C → F 8, Simp 10. ~H → E 8, Simp 11. F ∨ E 4, 9, 10, CD 12. E 11, 6, DS 106 SOLUTIONS MANUAL 18. 1. ~N • ~M 2. ~P → N 3. ~N → Z 4. (Z • ~~P) → K ∴ K • ~M 5. ~N 1, Simp 6. ~~P 2, 5, MT 7. Z 3, 5, MP 8. Z • ~~P 6, 7, Conj 9. K 4, 8, MP 10. ~M 1, Simp 11. K • ~M 9, 10, Conj *19. 1. A ∨ D 2. ~D 3. (C ∨ A) → ~E ∴ ~E 4. A 1, 2, DS 5. C ∨ A 4, Add 6. ~E 3, 5, MP 20. 1. (C → Q) • (~L → ~R) 2. (S → C) • (~N → ~L) 3. ~Q • J 4. ~Q → (S ∨ ~N) ∴ ~R 5. ~Q 3, Simp 6. S ∨ ~N 4, 5, MP 7. S → C 2, Simp 8. ~N → ~L 2, Simp 9. C ∨ ~L 6, 7, 8, CD 10. C → Q 1, Simp 11. ~L → ~R 1, Simp 12. Q ∨ ~R 9, 10, 11, CD 13. ~R 5, 12, DS 21. 1. ~(Z ∨ Y) → ~W 2. ~U → ~(Z ∨ Y) 3. (~U → ~W) → (T → S) 4. S → (R ∨ P) 5. [T → (R ∨ P)] → [(~R ∨ K) • ~K] ∴ ~R 6. ~U → ~W 2, 1, HS 7. T → S 3, 6, MP 8. T → (R ∨ P) 7, 4, HS 9. (~R ∨ K) • ~K 5, 8, MP 10. ~R ∨ K 9, Simp 11. ~K 9, Simp 12. ~R 10, 11, DS *22. 1. ~A 2. [~A ∨ (B • C)] → (D → ~E) 3. ~E → ~F 4. (D → ~F) → G ∴ (G • ~A) ∨ ~H 5. ~A ∨ (B • C) 1, Add 6. D → ~E 2, 5, MP 7. D → ~F 6, 3, HS 8. G 4, 7, MP 9. G • ~A 8, 1, Conj 10. (G • ~A) ∨ ~H 9, Add CHAPTER 8: STATEMENT LOGIC: PROOFS 107 23. 1. (S ∨ U) • ~U 2. S → [T • (F ∨ G)] 3. [T ∨ (J • P)] → (~B • E) ∴ S • ~B 4. ~U 1, Simp 5. S ∨ U 1, Simp 6. S 4, 5, DS 7. T • (F ∨ G) 2, 6, MP 8. T 7, Simp 9. T ∨ (J • P) 8, Add 10. ~B • E 9, 3, MP 11. ~B 10, Simp 12. S • ~B 6, 11, Conj 24. 1. ~X → (~Y → ~Z) 2. X ∨ (W → U) 3. ~Y ∨ W 4. ~X • T 5. (~Z ∨ U) → ~S ∴ (R ∨ ~S) • T 6. ~X 4, Simp 7. ~Y → ~Z 1, 6, MP 8. W → U 2, 6, DS 9. ~Z ∨ U 3, 7, 8, CD 10. ~S 5, 9, MP 11. R ∨ ~S 10, Add 12. T 4, Simp 13. (R ∨ ~S) • T 11, 12, Conj *25. 1. (D ∨ C) → (F ∨ H) 2. (H • G) → (F ∨ E) 3. (D ∨ B) → (~F → G) 4. (F ∨ D) • (~F • A) ∴ E 5. F ∨ D 4, Simp 6. ~F • A 4, Simp 7. ~F 6, Simp 8. D 5, 7, DS 9. D ∨ C 8, Add 10. F ∨ H 1, 9, MP 11. D ∨ B 8, Add 12. ~F → G 3, 11, MP 13. G 12, 7, MP 14. H 10, 7, DS 15. H • G 13, 14, Conj 16. F ∨ E 2, 15, MP 17. E 16, 7, DS Part E: English Arguments *1. 1. S 2. S → U 3. U → A 4. A → K ∴ K 5. S → A 2, 3, HS 6. S → K 5, 4, HS 7. K 6, 1, MP 108 SOLUTIONS MANUAL 2. 1. W → E 2. E → H 3. H → ~G 4. ~G → ~X ∴ W → ~X 5. W → H 1, 2, HS 6. W → ~G 5, 3, HS 7. W → ~X 6, 4, HS 3. 1. P ∨ (H ∨ G) 2. G → O 3. ~P • ~O ∴ H 4. ~P 3, Simp 5. H ∨ G 1, 4, DS 6. ~O 3, Simp 7. ~G 2, 6, MT 8. H 5, 7, DS *4. 1. E 2. E → (D → C) 3. (F ∨ V) → D 4. S → F 5. A • S ∴ C 6. S 5, Simp 7. F 4, 6, MP 8. F ∨ V 7, Add 9. D 3, 8, MP 10. D → C 2, 1, MP 11. C 10, 9, MP 5. 1. E → (A → P) 2. ~(A → P) ∴ ~E 3. ~E 1, 2, MT 6. 1. ~S → (M ∨ V) 2. S → (A → R) 3. ~(A → R) 4, M → (C • I) 5. ~(C • I) 6. V → G ∴ G 7. ~M 4, 5, MT 8. ~S 2, 3, MT 9. M ∨ V 8, 1, MP 10. V 9, 7, DS 11. G 10, 6, MP *7. 1. P 2. P → A 3. A → (E ∨ F) 4. ~E ∴ F 5. P → (E ∨ F) 2, 3, HS 6. E ∨ F 5, 1, MP 7. F 6, 4, DS

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