Symbolic Logic 5E: 3.1, IV

“Construct a formal proof of validity for each of the following arguments, using the abbreviations suggested”

1.

  1. (A ∨ G)→S
  2. A ∧ T …Therefore, S
  3. A (2,SIMP)
  4. A ∨ G (3,ADD)
  5. S (4,1,MP)

2.

  1. A ∨ ¬I
  2. D→I
  3. ¬A
  4. (¬D ∧ ¬I)→W …Therefore, W
  5. ¬I (3,1,DS)
  6. ¬D (5,2,MT)
  7. ¬D ∧ ¬I (6,5,CONJ)
  8. W (7,4,MP)

3.

  1. S→P
  2. C→¬F
  3. I→F
  4. O→¬P
  5. O ∨ C …Therefore, ¬S ∨ ¬I
  6. (O→¬P) ∧ (C→¬F) (4,2,CONJ)
  7. ¬P ∨ ¬F (6,5,CD)
  8. (I→F) ∧ (S→P) (3,1,CONJ)
  9. ¬S ∨ ¬I (8,7,DD)

4.

  1. C→N
  2. N→I
  3. I→S
  4. (C→S)→(N→C)
  5. ¬C …Therefore, ¬N
  6. C→I (1,2,HS)
  7. C→S (6,3,HS)
  8. N→C (7,4,MP)
  9. ¬N (5,8,MT)

5.

  1. (¬K ∧ P)→(B ∨ R)
  2. ¬K→(B→D)
  3. K ∨ (R→E)
  4. ¬K ∧ P …Therefore, D ∨ E
  5. ¬K (4,SIMP)
  6. B→D (2,5,MP)
  7. R→E (3,5,DS)
  8. (B→D) ∧ (R→E) (6,7,CONJ)
  9. B ∨ R (1,4,MP)
  10. D ∨ E (8,9,CD)

6.

  1. (A→B) ∧ (B→¬C)
  2. C→¬D
  3. B→E
  4. ¬D→F
  5. ¬E ∨ ¬F …Therefore, ¬A ∨ ¬C
  6. (B→E) ∧ (¬D→F) (4,3,CONJ)
  7. (¬B ∨ ¬¬D) (5,6,DD)
  8. A→B (1,SIMP)
  9. (A→B) ∧ (C→¬D) (8,2,CONJ)
  10. ¬A ∨ ¬C (9,7,DD)

7.

  1. (G ∨ H)→¬I
  2. I ∨ H
  3. (H ∨ ¬G)→J
  4. G …Therefore, J ∨ ¬H
  5. G ∨ H (4,ADD)
  6. ¬I (5,1,MP)
  7. H (6,2,DS)
  8. H ∨ ¬G (7,ADD)
  9. J (8,3,MP)
  10. J ∨ ¬H (9,ADD)

8.

  1. (R→P) ∧ (¬P→M)
  2. (M→D) ∧ (D→R)
  3. (¬M ∨ ¬R)→(¬P ∨ ¬D)
  4. ¬M …Therefore, ¬R ∨ ¬M
  5. ¬M ∨ ¬R (3,ADD)
  6. ¬R ∨ ¬M (5,COMM)

9.

  1.  V→F
  2. V ∨ (P→Q)
  3. M ∨ (R→C)
  4. M→F
  5. (¬V ∧ ¬M)→(R ∨ P)
  6. ¬F …Therefore, C ∨ Q
  7. ¬V (6,1,MT)
  8. ¬M (6,4,MT)
  9. ¬V ∧ ¬M (7,8,CONJ)
  10. R ∨ P (9,5,MP)
  11. P→Q (7,2,DS)
  12. R→C (8,3,DS)
  13. (R→C) ∧ (P→Q) (11,12,CONJ)
  14. C ∨ Q (10,13, CD)

10

  1. T ∨ (E→D)
  2. T→C
  3. (E→G)→(D→I)
  4. (¬T  ∨ ¬C)→(D→G)
  5. ¬C
  6. ¬I ∨ ¬G …Therefore, ¬D ∨ ¬E
  7. ¬T (2,5,MT)
  8. E→D (7,1,DS)
  9. ¬T  ∨ ¬C (7,ADD)
  10. D→G (9,4,MP)
  11. E→G (8,10,HS)
  12. D→I (3,11,MP)
  13. (D→I) ∧ (E→G ) (12,11,CONJ)
  14. ¬D ∨ ¬E (13,6,DD)
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