Category Archives: Solution Sets

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

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Symbolic Logic 5E: 3.4

“Give conditional proofs of validity for Exercises 21*, 22, 23, 24, and 25 on pages 47-48.”

21. In back of book.

22.

  1. (T→E) ∧ (A→L) …Therefore, (T ∧ A)→(E ∧ L)
  2. T ∧ A (CP)
  3. T (2,SIMP)
  4. T→E (1,SIMP)
  5. E (3,4,MP)
  6. A→L (1,SIMP)
  7. A (2,SIMP)
  8. L (7,6,MP)
  9. E ∧ L (5,8,CONJ)
  10. (T ∧ A)→(E ∧ L) (2-9,CP)

23.

  1. E→S
  2. E→(S→N)
  3. S→(N→F) …Therefore, E→F
  4. E (CP)
  5. S (4,1,MP)
  6. S→N (2,4,MP)
  7. N (5,6,MP)
  8. N→F (5,3,MP)
  9. F (7,8,MP)
  10. E→F (4-9,CP)

24.

  1. A→(B ∨ C)
  2. E→(C ∨ P)
  3. ¬C …Therefore, ¬(B ∨ P)→¬(A ∨ E)
  4. ¬(B ∨ P) (CP)
  5. ¬B ∧ ¬P (4,DeM)
  6. ¬B (5,SIMP)
  7. ¬B ∧ ¬C (6,3,CONJ)
  8. ¬(B ∨ C) (7,DeM)
  9. ¬A (8,1,MT)
  10. ¬P (5,SIMP)
  11. ¬P ∧ ¬C (3,9,CONJ)
  12. ¬(E ∨ P) (11,DeM)
  13. ¬E (12,2,MT)
  14. ¬A ∧ ¬E (9,13,CONJ)
  15. ¬(A ∨ E) (14,DeM)
  16. ¬(B ∨ P)→¬(A ∨ E) (4-15,CP)

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Symbolic Logic 5E: 3.3

“Prove the validity of each of the following arguments by the method of assigning truth values”

1. In back of book.

2. E=T ; F=T ; G=F ; H=F ; I=F

3. J=T ; K=F ; L=F ; M=F ; N=F

4. O=F ; P=T ; Q=T ; R=T ; S=T

5. In back of book.

6. X=F ; Y=T ; Z=F

7. A=F ; B=F ; C=T ; D=T ;  E=T ; F=T ; G=T ; H=F

8. I=T ; J=T ; K=F ; L=F; M=T ; N=F ; M=F

9. P=T ; Q=T ; R=F ; S=T ; T=T ; U=T ; V=F

10. In back of book.

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Symbolic Logic 5E: 3.2, III, Second Half

Note: I cheated and used conditional proof (CP) for problems 27-29. I don’t think this is such an egregious offense, as CP is introduced a mere four pages later. The point of doing these problems pre-CP is to see how helpful CP is. So save yourself some time and trust me: CP makes solving proofs much, much easier!

16.

  1. A→(B→C)
  2. C→(D ∧ E) …Therefore, A→(B→D)
  3. ¬A ∨ (B→C) (1,IMP)
  4. ¬A ∨ (¬B ∨ C) (3,IMP)
  5. (¬A ∨ ¬B) ∧ (¬A ∨ C) (4,DIST)
  6. ¬A ∨ ¬B (5,SIMP)
  7. ¬(A ∧ B) (6,DeM)
  8. ¬(A ∧ B) ∨ D (7,ADD)
  9. (A ∧ B)→D (8,IMP)
  10. A→(B→D) (9,EXP)

17.

  1. E→F
  2. G→F …Therefore, (E ∨ G)→F
  3. ¬E ∨ F (1,IMP)
  4. ¬G ∨ F (2,IMP)
  5. (¬E ∨ F) ∧ (¬G ∨ F) (3,4,CONJ)
  6. F ∨ (¬E ∧ ¬G) (5,DIST)
  7. (¬E ∧ ¬G) ∨ F (6,COMM)
  8. ¬(¬E ∧ ¬G)→F (7,IMP)
  9. (E ∨ G)→F (8,DeM)

18.

  1. [(H ∧ I)→J] ∧ [¬K→(I ∧ ¬J)] …Therefore, H→K
  2. (H ∧ I)→J (1,SIMP)
  3. ¬(H ∧ I) ∨ J (2,IMP)
  4. (¬H ∨ ¬I) ∨ J (3,DeM)
  5. ¬H ∨(¬I ∨ J) (4,ASSOC)
  6. H→(¬I ∨ J) (5,IMP)
  7. ¬K→(I ∧ ¬J) (1,SIMP)
  8. ¬(I ∧ ¬J)→K (7,TRANS)
  9. (¬I ∨ J)→K (8,DeM)
  10. H→K (6,9,HS)

19.

  1. [L ∧ (M ∨ N)]→(M ∧ N) …Therefore, L→(M→N)
  2. L→[(M ∨ N)→(M ∧ N)] (1,EXP)
  3. L→[¬(M ∨ N) ∨ (M ∧ N)] (2,IMP)
  4. L→[(¬M ∧ ¬N) ∨ (M ∧ N)] (3,DeM)
  5. L→(M≡N) (4,EQUIV)
  6. L→[(M→N) ∧ (N→M)] (5,EQUIV)
  7. ¬L ∨ [(M→N) ∧ (N→M)] (6,IMP)
  8. [¬L ∨ (M→N)] ∧ [¬L ∨ (N→M)] (7,DIST)
  9. ¬L ∨ (M→N) (8,SIMP)
  10. L→(M→N) (9,IMP)

20. In back of book.

21.

  1. S→(T ∧U)
  2. (T ∨ U)→V …Therefore, S→V
  3. ¬S ∨ (T ∧U) (1,IMP)
  4. (¬S ∨ T) ∧ (¬S ∨ U) (3,DIST)
  5. ¬(T ∨ U) ∨ V (2,IMP)
  6. (¬T ∧ ¬U) ∨ V (5,DeM)
  7. (V ∨ ¬T) ∧ (V ∨ ¬U) (6,DIST)
  8. ¬S ∨ T (4,SIMP)
  9. S→T (8,IMP)
  10. V ∨ ¬T (7,SIMP)
  11. ¬T ∨ V(10,COMM)
  12. T→V (11,IMP)
  13. S→V (9,12,HS)

22.

  1. ¬W ∨ [(X→Y) ∧ (Z→Y)]
  2. W ∧ (X ∨ V) …Therefore, Y
  3. W (2,SIMP)
  4. (X→Y) ∧ (Z→Y) (3,1,DS)
  5. X ∨ V (2,SIMP)
  6. Y ∨ Y (5,4,CD)
  7. Y (6,TAUT)

23.

  1. (A ∨ B)→(C ∧ D)
  2. ¬A→(E→¬E)
  3. ¬C …Therefore, ¬E
  4. A ∨ (E→¬E) (2,IMP)
  5. A ∨ (¬E ∨ ¬E) (4,IMP)
  6. A ∨ ¬E (5,TAUT)
  7. ¬C ∨ ¬D (3,ADD)
  8. ¬(C ∧ D) (7,DeM)
  9. ¬(A ∨ B) (8,1,MT)
  10. ¬A ∧ ¬B (9,DeM)
  11. ¬A (10,IMP)
  12. ¬E (11,6,DS)

24.

  1. (F→G) ∧ (H→I)
  2. F ∨ H
  3. (F→¬I) ∧ (H→G) …Therefore, G≡¬I
  4. G ∨ I (2,1,CD)
  5. I ∨ G (4,COMM)
  6.  ¬I→G (5,IMP)
  7. ¬G ∨ ¬I (2,3,CD)
  8. G→¬I (7,IMP)
  9. (G→¬I) ∧ (¬I→G) (8,6,CONJ)
  10. G≡¬I (9,EQUIV)

25. In back of book.

26.

  1. Q ∨ (R ∧ S)
  2. (Q→T) ∧ (T→S) …Therefore, S
  3. Q→T (2,IMP)
  4. T→S (2,IMP)
  5. Q→S (3,4,HS)
  6. (Q ∨ R) ∧ (Q ∨ S) (1,DIST)
  7. Q ∨ S (6,SIMP)
  8. ¬S→¬Q (5,TRANS)
  9. ¬Q→S (7,IMP)
  10. ¬S→S (6,9,HS)
  11. S ∨ S (10,IMP)
  12. S (11,TAUT)

27.

  1. (U→V) ∧ (W→X) …Therefore, (U ∨ W)→(V ∨ X)
  2. U→V (CP)
  3. V ∨ X (2,1,CD)
  4. (U ∨ W)→(V ∨ X) (2-3,CP)

28.

  1. (Y→Z) ∧(A→B) …Therefore, (Y ∧ A)→(Z ∧ B)
  2. Y ∧ A (CP)
  3. Y→Z (1,SIMP)
  4. Y (2,SIMP)
  5. Z (4,3,MP)
  6. A→B (1,SIMP)
  7. A (2,SIMP)
  8. B (7,6,MP)
  9. Z ∧ B (5,8,CONJ)
  10. (Y ∧ A)→(Z ∧ B) (2-9,CP)

29.

  1. (C→D) ∧ (E→F)
  2. G→(C ∨ E) …Therefore, G→(D ∨ F)
  3. G (CP)
  4. C ∨ E (3,2,MP)
  5. D ∨ F (4,1,CD)
  6. G→(D ∨ F) (3-5,CP)

30.

  1. (H→I) ∧ (J→K)
  2. H ∨ J
  3. (H→¬K) ∧ (J→¬I)
  4. (I ∧ ¬K)→L
  5. K→(I ∨ M) …Therefore, L ∨ M
  6. I ∨ K (1,2,CD)
  7. ¬K ∨ ¬I (2,3,CD)
  8. ¬I→K (6,IMP)
  9. K→¬I (7,IMP)
  10. ¬I→¬I (8,9,HS)
  11. I ∨ ¬I (10,IMP)
  12. (¬K ∧ I)→L (4,COM)
  13. ¬K→(I→L) (12,EXP)
  14. I→¬K (9,TRANS)
  15. I→(I→L) (14,13,HS)
  16. (I ∧ I)→L (15,EXP)
  17. I→L (16,TAUT)
  18. ¬I→(I ∨ M) (8,5,HS)
  19. I ∨ (I ∨ M) (18,IMP)
  20. (I ∨ I) ∨ M (19,ASSOC)
  21. I ∨ M (20,TAUT)
  22. ¬I→M (21,IMP)
  23. (I→L) ∧ (¬I→M) (17,22,CONJ)
  24. L ∨ M (23,11,CD)

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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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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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Symbolic Logic 5E: 3.2, I

“For each of the following arguments, state the Rule of Inference by which its conclusion follows from its premiss”

  1. Commutation.
  2. Material Implication.
  3. Transposition.
  4. De Morgan’s Theorem.
  5. Tautology.
  6. Association.
  7. Exportation.
  8. Material Equivalence.
  9. Distribution.
  10. Commutation
  11. De Morgan’s Theorem.
  12. Exportation.
  13. Association.
  14. Material Equivalence.
  15. Distribution.
  16. Double Negation.
  17. Material Implication.
  18. Transposition.
  19. Exportation.
  20. Exportation.

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