Symbolic Logic 5E: 3.5

“For each of the following arguments, construct both a formal proof of validity and an indirect proof and compare their length”

1.

Formal Proof

1. A ∨(B ∧ C)
2. A→C …Therefore, C
3. ¬A→(B ∧ C) (1,IMP)
4. ¬C→¬A (2,TRANS)
5. ¬C→(B ∧ C) (4,3,HS)
6. C ∨ (B ∧ C) (5,IMP)
7. (C ∨ B) ∧ (C ∧ C) (6,DIST)
8. C ∧ C (7,SIMP)
9. C (8,TAUT)

Indirect Proof

1. A ∨(B ∧ C)
2. A→C …Therefore, C
3. ¬C (IP)
4. ¬A (3,2,MT)
5. B ∧ C (4,1,DS)
6. C (5,SIMP)
7. C ∧ ¬C (6,3,CONJ)

2.

Formal Proof

 

Indirect Proof

1. (D ∨ E)→(F→G)
2. (¬G ∨ H)→(D ∧ F) …Therefore, G
3. ¬G (IP)
4. ¬G ∨ H (3,ADD)
5. D ∧ F (4,2,MP)
6. D (5,SIMP)
7. D ∨ E (6,ADD)
8. F→G (8,1,MP)
9. F (5,SIMP)
10. G (9,8,MP)
11. G ∧ ¬G (10,3,CONJ)

3. In back of book.

4.

Formal Proof

 

Indirect Proof

1. (M ∨ N)→(O ∧ P)
2. (O ∨ Q)→(¬R ∧ S)
3. (R ∨ T)→(M ∧ U) …Therefore, ¬R
4. R (IP)
5. R ∨ T (4,ADD)
6. M ∧ U (5,3,MP)
7. M (6,SIMP)
8. M ∨ N (7,ADD)
9. O ∧ P (8,1,MP)
10. O (9,SIMP)
11. O ∨ Q (10,ADD)
12. ¬R ∧ S (11,2,MP)
13. ¬R (12,SIMP)
14. ¬R ∧ R (13,4,CONJ)

5. In back of book.

∧ = And
∨ = Or
→ = If…then
¬  = Not (negation)
≡   = Logical Equivalence