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Exercise Answers Patrick J. Hurley, A Concise Introduction to Logic, 11th ed. (2012) Keith Burgess-Jackson 15 November 2011 Section 6.3, Part I: 1. Tautologous 2. Contingent 3. Self-contradictory 4. Contingent 5. Self-contradictory 6. Tautologous 7. Tautologous 8. Contingent 9. Contingent 10. Self-contradictory 11. Self-contradictory 12. Tautologous 13. Self-contradictory 14. Tautologous 15. Contingent Section 6.3, Part II: 1. Logically equivalent; consistent 2. Inconsistent; contraries 3. Consistent; subcontraries 4. Contradictory; inconsistent 5. Logically equivalent; consistent 6. Contradictory; inconsistent 7. Consistent 8. Inconsistent; contraries 9. Logically equivalent; consistent 10. Inconsistent; contraries 11. Contradictory; inconsistent 12. Consistent; superalternates 13. Logically equivalent; consistent 14. Inconsistent; contraries 15. Contradictory; inconsistent Section 6.5, Part II: 1. Valid (one row required) 2. Invalid (one row) 3. Valid (one row) 4. Invalid (two rows) 1

5. Valid (one row) 6. Invalid (one row) 7. Valid (one row) 8. Invalid (one row) 9. Valid (two rows) 10. Invalid (two rows) 11. Valid (one row) 12. Valid (one row) 13. Valid (one row) 14. Invalid (three rows) 15. Invalid (three rows) Section 6.5, Part III: 1. Inconsistent (one row required) 2. Consistent (one row) 3. Inconsistent (one row) 4. Consistent (two rows; Hurley used three rows) 5. Consistent (three rows) 6. Inconsistent (one row) 7. Inconsistent (one row) 8. Consistent (three rows) 9. Inconsistent (three rows) 10. Consistent (three rows) Section 7.4, Part III: 2. 1. T  (F v F) 2. ~(F  F) 3. ~F 4. T  F 5. ~T

/ ~T 2, Taut 1, Taut 3, 4, MT

3. 1. 2. 3. 4. 5. 6.

/ G  ~H 1, Trans 2, 3, HS 4, Trans 5, DN

G  E H  ~E ~E  ~G H  ~G ~~G  ~H G  ~H

3. Alternative (shorter) proof 1. G  E 2. H  ~E

/ G  ~H

2

3. ~~E  ~H 4. E  ~H 5. G  ~H

2, Trans 3, DN 1, 4, HS

5. 1. 2. 3. 4.

~N v P (N  P)  T N  P T

/ T 1, Impl 2, 3, MP

6. 1. 2. 3. 4. 5.

F  B B  (B  J) (B  B)  J B  J F  J

/ F  J 2, Exp 3, Taut 1, 4, HS

8. 1. Q  (F  A) 2. R  (A  F) 3. Q  R 4. Q 5. F  A 6. R  Q 7. R 8. A  F 9. (F  A)  (A  F) 10. F  A

/ F  A 3, Simp 1, 4, MP 3, Com 6, Simp 2, 7, MP 5, 8, Conj 9, Equiv

9. 1. 2. 3. 4. 5. 6. 7.

T  (~T v G) ~G ~T v (~T v G) (~T v ~T) v G ~T v G G v ~T ~T

/ ~T 1, Impl 3, Assoc 4, Taut 5, Com 2, 6, DS

11. 1. 2. 3. 4.

(J  R)  H (R  H)  M ~(P v ~J) ~P  ~~J

/ M  ~P 3, DM

3

5. ~P 6. J  (R  H) 7. ~~J  ~P 8. ~~J 9. J 10. R  H 11. M 12. M  ~P

4, Simp 1, Exp 4, Com 7, Simp 8, DN 6, 9, MP 2, 10, MP 5, 11, Conj

12. 1. 2. 3. 4.

/ S  T 1, Add 2, Com 3, Impl

T T v ~S ~S v T S  T

Section 7.5, Part I: 2. 1. F  E 2. (F  E)  R 3. F 4. E 5. F  E 6. R 7. F  R

/ F  R ACP 1, 3, MP 3, 4, Conj 2, 5, MP 3-6, CP

3. 1. G  T 2. (T v S)  K 3. G 4. T 5. T v S 6. K 7. G  K

/ G  K ACP 1, 3, MP 4, Add 2, 5, MP 3-6, CP

5. 1. A  ~(A v E) 2. A 3. ~(A v E) 4. ~A  ~E 5. ~A 6. A v F 7. F 8. A  F

/ A  F ACP 1, 2, MP 3, DM 4, Simp 2, Add 5, 6, DS 2-7, CP

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5. Direct proof 1. A  ~(A v E) 2. ~A v ~(A v E) 3. ~A v (~A  ~E) 4. (~A v ~A)  (~A v ~E) 5. ~A v ~A 6. ~A 7. ~A v F 8. A  F

/ A  F 1, Impl 2, DM 3, Dist 4, Simp 5, Taut 6, Add 7, Impl

6. 1. J  (K  L) 2. J  (M  L) 3. ~L 4. J 5. K  L 6. ~K 7. M  L 8. ~M 9. ~K  ~M 10. ~(K v M) 11. J  ~(K v M)

/ J  ACP 1, 4, 3, 5, 2, 4, 3, 7, 6, 8, 9, DM 4-10,

~(K v M) MP MT MP MT Conj CP

8. (Indirect truth table proves validity in one line) 1. P  (Q  R) 2. (P  R)  (S • T) 3. Q  R / T 4. P ACP 5. Q  R 1, 4, MP 6. R  Q 5, Com 7. R  Q 6, DN 8. R  Q 7, Impl 9. R  R 3, 8, HS 10. R  R 9, Impl 11. R  R 10, DN 12. R 11, Taut 13. P  R 4-12, CP 14. S • T 2, 13, MP 15. T • S 14, Com 16. T 15, Simp 9.

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1. H  (I  N) 2. (H  ~I)  (M v N) 3. ~N 4. H 5. I  N 6. ~I 7. H  ~I 8. M v N 9. N v M 10. M

/ M ACP 1, 4, MP 3, 5, MT 4-6, CP 2, 7, MP 8, Com 3, 9, DS

11. 1. M  (K  L) 2. (L v N)  J 3. M 4. K  L 5. K 6. L 7. L v N 8. J 9. K  J 10. M  (K  J)

/ M  (K  J) ACP 1, 3, MP ACP 4, 5, MP 6, Add 2, 7, MP 5-8, CP 3-9, CP

12. 1. F  (G  H) 2. A  F 3. A  (G  H) 4. ~A v (G  H) 5. (~A v G)  (~A v H) 6. (~A v H)  (~A v G) 7. ~A v H 8. A  H 9. (A  F)  (A  H)

/ (A  F)  (A  H) ACP 1, 2, HS 3, Impl 4, Dist 5, Com 6, Simp 7, Impl 2-8, CP

14. 1. 2. 3. 4. 5. 6.

(F  G)  H F  G [(F  G)  H]  [H  (F  G)] (F  G)  H [H  (F  G)]  [(F  G)  H] H  (F  G) 7. F 8. G

/ F  H 1, Equiv 3, Simp 3, Com 5, Simp ACP 2, 7, MP

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9. F  G 10. H 11. F  H 12. H 13. F  G 14. F 15. H  F 16. (F  H)  (H  F) 17. F  H

7, 8, Conj 4, 9, MP 7-10, CP ACP 6, 12, MP 13, Simp 12-14, CP 11, 15, Conj 16, Equiv

15. 1. C  (D v ~E) 2. E  (D  F) 3. C 4. D v ~E 5. ~E v D 6. E  D 7. E 8. D 9. D  F 10. F 11. E  F 12. C  (E  F)

/ C  (E  F) ACP 1, 3, MP 4, Com 5, Impl ACP 6, 7, MP 2, 7, MP 8, 9, MP 7-10, CP 3-11, CP

17. 1. N  (O  P) 2. Q  (R  S) 3. P  Q 4. N 5. O  P 6. P  O 7. P 8. Q 9. R  S 10. S  R 11. S 12. N  S 13. (P  Q)  (N  S)

/ (P  Q)  (N  S) ACP ACP 1, 4, MP 5, Com 6, Simp 3, 7, MP 2, 8, MP 9, Com 10, Simp 4-11, CP 3-12, CP

18. 1. E  (F  G) 2. H  (G  I) 3. (F  I)  (J v ~H)

/ (E  H)  J

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4. E  H 5. E 6. F  G 7. H  E 8. H 9. G  I 10. F  I 11. J v ~H 12. ~H v J 13. H  J 14. J 15. (E  H)  J

ACP 4, Simp 1, 5, MP 4, Com 7, Simp 2, 8, MP 6, 9, HS 3, 10, MP 11, Com 12, Impl 8, 13, MP 4-14, CP

20. 1. A  [B  (C  ~D)] 2. (B v E)  (D v E) 3. A  B 4. A 5. B  (C  ~D) 6. B  A 7. B 8. C  ~D 9. C 10. B v E 11. D v E 12. ~D  C 13. ~D 14. E 15. C  E 16. (A  B)  (C  E)

/ (A  B)  (C  E) ACP 3, Simp 1, 4, MP 3, Com 6, Simp 5, 7, MP 8, Simp 7, Add 2, 10, MP 8, Com 12, Simp 11, 13, DS 9, 14, Conj 3-15, CP

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