Symbolic Logic 5E: 3.1, III

“Construct a formal proof of validity for each of the following arguments”

1.

  1. A→B
  2. C→D
  3. (¬B ∨ ¬D) ∧ (¬A ∨ ¬B) …Therefore, ¬A ∨ ¬C
  4. (A→B) ∧ (C→D) (1,2,CONJ)
  5. ¬B ∨ ¬D (3,SIMP)
  6. ¬A ∨ ¬C (5,4,DD)

2.

  1. E→(F ∧ ¬G)
  2. (F ∨ G)→H
  3. E …Therefore, H
  4. F ∧ ¬G (3,1,MP)
  5. F (4,SIMP)
  6. F ∨ G (5,ADD)
  7. H (6,2,MP)

3.

  1. J→K
  2. J ∨ (K ∨ ¬L)
  3. ¬K …Therefore, ¬L ∧ ¬K
  4. ¬J (3,1,MT)
  5. K ∨ ¬L (2,4,DS)
  6. ¬L (3,5,DS)
  7. ¬L ∧ ¬K (6,3,CONJ)

4.

  1. M→N
  2. N→O
  3. (M→O)→(N→P)
  4. (M→P)→Q …Therefore, Q
  5. M→O (1,2,HS)
  6. N→P (5,3,MP)
  7. M→P (6,1,HS)
  8. Q (7,4,MP)

5.

  1. (R→¬S) ∧ (T→¬U)
  2. (V→¬W) ∧ (X→¬Y)
  3. (T→W) ∧ (U→S)
  4. V ∨ R …Therefore, ¬T ∨ ¬U
  5. R→¬S (1,SIMP)
  6. V→¬W (2,SIMP)
  7. (R→¬S) ∧ (V→¬W) (5,6,CONJ)
  8. ¬S ∨ ¬W (7,4,CD)
  9. ¬T ∨ ¬U (8,3,DD)

6.

  1. A→(B ∧ C)
  2. ¬A→[(D→E) ∧ (F→G)]
  3. (B ∧ C) ∨ [(¬A→D) ∧ (¬A→F)]
  4. ¬(B ∧ C) ∧ ¬(G ∧ D) …Therefore, E ∨ G
  5. ¬(B ∧ C) (4,SIMP)
  6. ¬A (5,1,MT)
  7. (¬A→D) ∧ (¬A→F) (6,2,DS)
  8. ¬A→D (7,SIMP)
  9. D (8,6,MP)
  10. D ∨ F (9,ADD)
  11. (D→E) ∧ (F→G) (2,8,MP)
  12. E ∨ G (11,10,CD)

7.

  1. (¬H ∨ I)→(J→K)
  2. (¬L ∧ ¬M)→(K→N)
  3. (H→L) ∧ (L→H)
  4. (¬L ∧ ¬M) ∧ ¬O …Therefore, J→N
  5. ¬L ∧ ¬M (4,SIMP)
  6. K→N (5,2,MP)
  7. H→L (3,SIMP)
  8. ¬L (5,SIMP)
  9. ¬H (8,7,MT)
  10. ¬H ∨ I (9,ADD)
  11. J→K (10,1,MP)
  12. J→N (11,6,HS)

8.

  1. (P→Q) ∧ (R→S)
  2. (Q→T) ∧ (S→U)
  3. (¬P→T) ∧ (¬Q→S)
  4. ¬T …Therefore, ¬R ∨ ¬Q
  5. Q→T (3,SIMP)
  6. ¬Q (4,5,MT)
  7. ¬Q ∨ ¬R (6,ADD)
  8. ¬R ∨ ¬Q (7,COMM)

9.

  1. V→W
  2. X→Y
  3. Z→W
  4. X→A
  5. W→X
  6. [(V→Y) ∧ (Z→A)]→(V ∨ Z) …Therefore, Y ∨ A
  7. V→ X (1,5,HS)
  8. V→Y (7,2,HS)
  9. Z→ X (3,5,HS)
  10. Z→ A (9,4,HS)
  11. [(V→Y) ∧ (Z→ A)] (8,10,CONJ)
  12. V ∨ Z (11,6,MP)
  13. Y ∨ A (12,6,CD)

10.

  1. (B ∨ C)→(D ∨ E)
  2. [(D ∨ E) ∨ F]→(G ∨ H)
  3. (G ∨ H)→¬D
  4. E→¬G
  5. B …Therefore, H
  6. B ∨ C (5,ADD)
  7. D ∨ E (6,1,MP)
  8. (D ∨ E) ∨ F (7,ADD)
  9. G ∨ H (8,2,MP)
  10. ¬D (9,3,MP)
  11. E (10,7,DS)
  12. ¬G (11,4,MP)
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