Symbolic Logic 5E: 3.2, II

“Each of the following is a formal proof of validity for the indicated argument. State the ‘justification’ for each line that is not a premiss”

1.

  1. (A ∨ B)→(C ∧ D)
  2. ¬C …Therefore, ¬B
  3. ¬C ∨ ¬D (2,ADD)
  4. ¬(C ∧ D) (3,DeM)
  5. ¬(A ∨ B) (1,4,MT)
  6. ¬A ∧ ¬B (5,DeM)
  7. ¬B ∧ ¬A (6,COMM)
  8. ¬B (7,ADD)

2.

  1. (E ∧ F) ∧ G
  2. (F≡G)→(H ∨ I) …Therefore, I ∨ H
  3. E ∧ (F ∧ G) (1,ASSOC)
  4. (F ∧ G) ∧ E (3,COMM)
  5. F ∧ G (4,SIMP)
  6. (F ∧ G) ∨ (¬F ∧ ¬G) (5,ADD)
  7. F≡G (6,EQUIV)
  8. H ∨ I (7,2,MP)
  9. I ∨ H (8,COMM)

3.

  1. (J ∧K)→L
  2. (J→L)→M
  3. ¬K ∨ N …Therefore, K→(M ∧ N)
  4. (K ∧ J)→L (1,COMM)
  5. K→(J→L) (4,EXP)
  6. K→M (5,2,HS)
  7. ¬K ∨ M (6,IMP)
  8. (¬K ∨ M) ∧ (¬K ∨ N) (3,8,CONJ)
  9. ¬K ∨ (M ∧ N) (8,DIST)
  10. K→(M ∧ N) (9,IMP)

4.

  1. (O→¬P) ∧ (P→Q)
  2. Q→O
  3. ¬R→P …Therefore, R
  4. ¬Q ∨ O (2,IMP)
  5. O ∨ ¬Q (4,COMM)
  6. (O→¬P) ∧ (¬Q→¬P) (1,TRANS)
  7. ¬P ∨ ¬P (5,6,CD)
  8. ¬P (7,TAUT)
  9. ¬¬R (8,3,MT)
  10. R (9,DN)

5.

  1. S→(T→U)
  2. U→¬U
  3. (V→S) ∧ (W→T) …Therefore, V→¬W
  4. (S ∧ T)→U (1,EXP)
  5. ¬U ∨ ¬U (2,IMP)
  6. ¬U (5,TAUT)
  7. ¬(S ∧ T) (4,6,MT)
  8. ¬S ∨ ¬T (7,DeM)
  9. ¬V ∨ ¬W (3,8,DD)
  10. V→¬W (9,IMP)

6.

  1. X→(Y→Z)
  2. X→(A→B)
  3. X ∧ (Y ∨ A)
  4. ¬Z …Therefore, B
  5. (X ∧ Y)→Z (1,EXP)
  6. (X ∧ A)→B (2,EXP)
  7. (X ∧ Y) ∨ (X ∧ A) (3,DIST)
  8. [(X ∧ Y)→Z] ∧ [(X ∧ A)→B] (5,6,CONJ)
  9. Z ∨ B (7,8,CD)
  10. B (9,4,DS)

7.

  1. C→(D→¬C)
  2. C≡D …Therefore, ¬C ∧ ¬D
  3. C→(¬¬C→¬D) (1,TRANS)
  4. C→(C→¬D) (3,DN)
  5. (C ∧ C)→¬D (4,EXP)
  6. C→¬D (5,TAUT)
  7. ¬C ∨ ¬D (6,IMP)
  8. ¬(C ∧ D) (7,DeM)
  9. (C ∧ D) ∨ (¬C ∧ ¬D) (2,EQUIV)
  10. ¬C ∧ ¬D (9,8,DS)

8.

  1. E ∧ (F ∨ G)
  2. (E ∧ G)→¬(H ∨ I)
  3. (¬H ∨ ¬I)→¬(E ∧ F) …Therefore, H≡I
  4. (E ∧ G)→(¬H ∧ ¬I) (2,DeM)
  5. ¬(H ∧ I)→¬(E ∧ F) (3,DeM)
  6. (E ∧ F)→(H ∧ I) (5,TRANS)
  7. [(E ∧ F)→(H ∧ I)] ∧ [(E ∧ G)→(¬H ∧ ¬I)] (6,4,CONJ)
  8. (E ∧ F) ∨ (E ∧ G) (1,DIST)
  9. (H ∧ I) ∨ (¬H ∧ ¬I) (7,8,CD)
  10. H≡I (9,EQUIV)

9.

  1. J ∨ (¬K ∨ J)
  2. K ∨ (¬J ∨ K) …Therefore, (J ∧ K) ∨ (¬J ∨ ¬K)
  3. (¬K ∨ J) ∨ J (1,COMM)
  4. ¬K ∨ (J ∨ J) (3,ASSOC)
  5. ¬K ∨ J (4,TAUT)
  6. K→J (5,IMP)
  7. (¬J ∨ K) ∨ K (2,COMM)
  8. ¬J ∨ (K ∨ K) (7,ASSOC)
  9. ¬J ∨ K (8,TAUT)
  10. J→K (9,IMP)
  11. (K→J) ∧ (J→K) (10,6,CONJ)
  12. J ≡K (11,EQUIV)
  13. (J ∧ K) ∨ (¬J ∨ ¬K) (12,EQUIV)

10.

  1. (L ∨ M) ∨ (N ∧ O)
  2. (¬L ∧ O) ∧ ¬(¬L ∧ M) …Therefore, ¬L ∧ N
  3. ¬L ∧ [O ∧ ¬(¬L ∧ M)] (2,ASSOC)
  4. ¬L (3,SIMP)
  5. L ∨ [ M ∨ (N ∧ O)] (1,ASSOC)
  6. M ∨ (N ∧ O) (5,4,DS)
  7. (M ∨ N) ∧ (M ∨ O) (6,DIST)
  8. M ∨ N (7,SIMP)
  9. ¬L ∧ (M ∨ N) (8,4,CONJ)
  10. (¬L ∧ M) ∨ (¬L ∧ N) (9,DIST)
  11. ¬(¬L ∧ M) ∧ (¬L ∧ O) (2,COMM)
  12. ¬(¬L ∧ M) (11,SIMP)
  13. ¬L ∧ N (12,10,DS)
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2 Comments

Filed under Solution Sets

2 responses to “Symbolic Logic 5E: 3.2, II

  1. regina domingo

    how can i construct a formal proof of validity to the following arguments:

    S>W
    W> ~L
    S
    D> ~I
    L V I V C
    C>B
    therefore B

    • I’ll post the proof in full, starting from the beginning.

      1. S->W
      2. W->~L
      3. S
      4. D->~I
      5. (L V I) V C
      6. C->B … Therefore, B
      7. S->~L (1-2, HS)
      8. ~L (7,3,MP)
      9. (L V I) ∧ (L V C) (5, DIST)
      10. L V C (9, SIMP)
      11. C (10,8, DS)
      12. B (11,6, MP)

      I put the parentheses around the first two symbols in premise 5, because disjunctions only include two variables. In other words, L V I V C doesn’t make sense; only (L V I) V C does. Be sure to use parentheses on disjunctions as appropriate.

      I’ll answer any further questions.

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