Symbolic Logic 5E: 3.2, III, First Half

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

1. In back of book.

2.

  1. C …Therefore, D→C
  2. C  ∨ ¬D (1,ADD)
  3. ¬D  ∨ C (2,COMM)
  4. D→C (3,IMP)

3.

  1. E→(F→G) …Therefore, F→(E→G)
  2. (E ∧ F)→G (1,EXP)
  3. (F ∧ E)→G (2,COMM)
  4. F→(E→G) (3,EXP)

4.

  1. H→(I ∧ J) …Therefore, H→I
  2. ¬H ∨ (I ∧ J) (1,IMP)
  3. [(¬H ∨ I) ∧ (¬H ∨ J)] (2,DIST)
  4. ¬H ∨ I (3,SIMP)
  5. H→I (4,IMP)

5. In back of book.

6.

  1. N→O …Therefore,  (N ∧ P)→O
  2. ¬N ∨ O (1,IMP)
  3. (¬N ∨O ) ∨ ¬P (2,ADD)
  4. ¬P ∨ (¬N ∨O ) (3,COMM)
  5. P→(¬N ∨O ) (4,IMP)
  6. P→(N→O) (5,IMP)
  7. (P ∧ N)→O (6,EXP)
  8. (N ∧ P)→O (7,COMM)

7.

  1. (Q ∨R)→S …Therefore, Q→S
  2. ¬(Q ∨R) ∨ S (1,IMP)
  3. (¬Q ∧ ¬R) ∨ S (2,DeM)
  4. (S ∨ ¬Q) ∧ (S ∨ ¬R) (3,DIST)
  5. S ∨ ¬Q (4,SIMP)
  6. ¬Q ∨ S (5,COMM)
  7. Q→S (6,IMP)

8.

  1. T→¬(U→V) …Therefore, T→U
  2. T→¬(¬U ∨ V) (1,IMP)
  3. T→(U ∧ ¬V) (2,DeM)
  4. ¬T ∨ (U ∧ ¬V) (3,IMP)
  5. (¬T ∨ U) ∧ (¬T ∨ ¬V) (4,DIST)
  6. ¬T ∨ U (5,SIMP)
  7. T→U (6,IMP)

9.

  1. W→(X ∧ ¬Y) …Therefore, W→(Y→X)
  2. ¬W ∨ (X ∧ ¬Y) (1,IMP)
  3. (¬W ∨ X) ∧ (¬W ∨ ¬Y) (2,DIST)
  4. ¬W ∨ X (3,SIMP)
  5. (¬W ∨ X) ∨ ¬Y (4,ADD)
  6. ¬W ∨ (X ∨ ¬Y) (5,ASSOC)
  7. W→(X ∨ ¬Y) (6,IMP)
  8. W→(¬Y ∨ X) (7,COMM)
  9. W→(Y→X) (8,IMP)

10. In back of book.

11.

  1. E→F
  2. E→G …Therefore, E→(F ∧ G)
  3. ¬E ∨ F (1,IMP)
  4. ¬E ∨ G (2,IMP)
  5. [(¬E ∨ F) ∧ (¬E ∨ G)] (3,4,CONJ)
  6. ¬E ∨ (F ∧ G) (5,DIST)
  7. E→(F ∧ G) (6,IMP)

12.

  1. H→(I ∨ J)
  2. ¬I …Therefore, H→J
  3. ¬H ∨ (I ∨ J) (1,IMP)
  4. (¬H ∨ I) ∧ (¬H ∨ J) (3,DIST)
  5. ¬H ∨ J (4,SIMP)
  6. H→J (5,IMP)

13.

  1. (K ∨ L)→¬(M ∧ N)
  2. (¬M ∨ ¬N)→(O≡P)
  3. (O≡P)→(Q ∧ R) …Therefore, (L ∨ K)→(R ∧ Q)
  4. (K ∨ L)→(¬M ∨ ¬N) (1,DeM)
  5. (K ∨ L)→(O≡P) (4,2,HS)
  6. (K ∨ L)→(Q ∧ R) (5,3,HS)
  7. (L ∨ K)→(Q ∧ R) (6,IMP)
  8. (L ∨ K)→(R ∧ Q) (7,IMP)

14.

  1. S→T
  2. S ∨ T …Therefore, T
  3. ¬T→¬S (1,TRANS)
  4. ¬S→T (2,IMP)
  5. ¬T→T (3,4,HS)
  6. T ∨ T (5,IMP)
  7. T (6,TAUT)

15. In back of book.

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