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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)

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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) / ~T

3. ~F 2, Taut

4. T F 1, Taut

5. ~T 3, 4, MT

3.

1. G E

2. H ~E / G ~H

3. ~E ~G 1, Trans

4. H ~G 2, 3, HS

5. ~~G ~H 4, Trans

6. G ~H 5, DN

3. Alternative (shorter) proof

1. G E

2. H ~E / G ~H

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3. ~~E ~H 2, Trans

4. E ~H 3, DN

5. G ~H 1, 4, HS

5. 1. ~N v P

2. (N P) T / T

3. N P 1, Impl 4. T 2, 3, MP

6.

1. F B

2. B (B J) / F J

3. (B B) J 2, Exp

4. B J 3, Taut

5. F J 1, 4, HS

8.

1. Q (F A)

2. R (A F)

3. Q R / F A

4. Q 3, Simp

5. F A 1, 4, MP

6. R Q 3, Com 7. R 6, Simp

8. A F 2, 7, MP

9. (F A) (A F) 5, 8, Conj

10. F A 9, Equiv

9.

1. T (~T v G) 2. ~G / ~T

3. ~T v (~T v G) 1, Impl

4. (~T v ~T) v G 3, Assoc

5. ~T v G 4, Taut

6. G v ~T 5, Com

7. ~T 2, 6, DS

11.

1. (J R) H

2. (R H) M

3. ~(P v ~J) / M ~P

4. ~P ~~J 3, DM

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5. ~P 4, Simp

6. J (R H) 1, Exp

7. ~~J ~P 4, Com 8. ~~J 7, Simp

9. J 8, DN

10. R H 6, 9, MP

11. M 2, 10, MP

12. M ~P 5, 11, Conj

12.

1. T / S T 2. T v ~S 1, Add

3. ~S v T 2, Com

4. S T 3, Impl

Section 7.5, Part I: 2.

1. F E

2. (F E) R / F R

3. F ACP

4. E 1, 3, MP

5. F E 3, 4, Conj 6. R 2, 5, MP

7. F R 3-6, CP

3.

1. G T

2. (T v S) K / G K

3. G ACP

4. T 1, 3, MP

5. T v S 4, Add

6. K 2, 5, MP

7. G K 3-6, CP

5.

1. A ~(A v E) / A F 2. A ACP

3. ~(A v E) 1, 2, MP

4. ~A ~E 3, DM 5. ~A 4, Simp

6. A v F 2, Add

7. F 5, 6, DS

8. A F 2-7, CP

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5. Direct proof

1. A ~(A v E) / A F

2. ~A v ~(A v E) 1, Impl

3. ~A v (~A ~E) 2, DM

4. (~A v ~A) (~A v ~E) 3, Dist 5. ~A v ~A 4, Simp

6. ~A 5, Taut

7. ~A v F 6, Add

8. A F 7, Impl

6.

1. J (K L)

2. J (M L)

3. ~L / J ~(K v M)

4. J ACP

5. K L 1, 4, MP

6. ~K 3, 5, MT

7. M L 2, 4, MP 8. ~M 3, 7, MT

9. ~K ~M 6, 8, Conj 10. ~(K v M) 9, DM

11. J ~(K v M) 4-10, 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 / M

4. H ACP

5. I N 1, 4, MP

6. ~I 3, 5, MT

7. H ~I 4-6, CP

8. M v N 2, 7, MP

9. N v M 8, Com

10. M 3, 9, DS

11.

1. M (K L)

2. (L v N) J / M (K J) 3. M ACP

4. K L 1, 3, MP

5. K ACP

6. L 4, 5, MP

7. L v N 6, Add

8. J 2, 7, MP

9. K J 5-8, CP

10. M (K J) 3-9, CP

12.

1. F (G H) / (A F) (A H)

2. A F ACP

3. A (G H) 1, 2, HS

4. ~A v (G H) 3, Impl

5. (~A v G) (~A v H) 4, Dist

6. (~A v H) (~A v G) 5, Com

7. ~A v H 6, Simp

8. A H 7, Impl

9. (A F) (A H) 2-8, CP

14.

1. (F G) H

2. F G / F H

3. [(F G) H] [H (F G)] 1, Equiv

4. (F G) H 3, Simp

5. [H (F G)] [(F G) H] 3, Com

6. H (F G) 5, Simp

7. F ACP

8. G 2, 7, MP

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9. F G 7, 8, Conj

10. H 4, 9, MP

11. F H 7-10, CP

12. H ACP

13. F G 6, 12, MP

14. F 13, Simp

15. H F 12-14, CP

16. (F H) (H F) 11, 15, Conj

17. F H 16, Equiv

15.

1. C (D v ~E)

2. E (D F) / C (E F)

3. C ACP

4. D v ~E 1, 3, MP

5. ~E v D 4, Com

6. E D 5, Impl

7. E ACP

8. D 6, 7, MP

9. D F 2, 7, MP

10. F 8, 9, MP

11. E F 7-10, CP

12. C (E F) 3-11, CP

17.

1. N (O P)

2. Q (R S) / (P Q) (N S)

3. P Q ACP

4. N ACP

5. O P 1, 4, MP

6. P O 5, Com

7. P 6, Simp

8. Q 3, 7, MP

9. R S 2, 8, MP

10. S R 9, Com

11. S 10, Simp

12. N S 4-11, CP

13. (P Q) (N S) 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 ACP

5. E 4, Simp

6. F G 1, 5, MP

7. H E 4, Com

8. H 7, Simp

9. G I 2, 8, MP

10. F I 6, 9, HS

11. J v ~H 3, 10, MP

12. ~H v J 11, Com

13. H J 12, Impl

14. J 8, 13, MP

15. (E H) J 4-14, CP

20.

1. A [B (C ~D)]

2. (B v E) (D v E) / (A B) (C E)

3. A B ACP

4. A 3, Simp

5. B (C ~D) 1, 4, MP

6. B A 3, Com

7. B 6, Simp

8. C ~D 5, 7, MP

9. C 8, Simp

10. B v E 7, Add

11. D v E 2, 10, MP

12. ~D C 8, Com

13. ~D 12, Simp

14. E 11, 13, DS

15. C E 9, 14, Conj

16. (A B) (C E) 3-15, CP