Symbolic Logic 5E: 2.3, III

“Use truth tables to determine the validity or invalidity of each of the following arguments”

1. Invalid.

  1. A→(B ∨ C)
  2. B …Therefore, A→¬C
A B C B ∨ C A→(B ∨ C) A→¬C ¬C

T

T

T

T

T

F

F

 X

T

T

F

T

T

F

T

T

F

F

F

F

T

T

T

F

T

T

T

F

F

F

F

F

F

T

T

T

F

F

T

T

T

T

F

F

T

T

T

T

T

F

F

T

F

T

T

T

T

2. Valid.

  1. A→(B ∨ C)
  2. ¬C …Therefore, ¬B→¬A
A B C B ∨ C A→(B ∨ C) ¬C ¬B→¬A ¬B ¬A

T

T

T

T

T

F

T

F

F

T

T

F

T

T

T

T

F

F

V

T

F

F

F

F

T

F

T

F

T

F

T

T

T

F

F

T

F

F

F

F

F

T

T

T

T

T

V

F

F

T

T

T

F

T

T

T

F

T

T

T

T

F

T

F

T

F

T

F

T

T

T

T

F

T

V

3. Valid.

  1. A→(B ∧ C)
  2. ¬B …Therefore, ¬A
A B ¬B C ¬A

B ∧ C

A→(B ∧ C)

T

T

F

T

F

T

T

T

T

F

F

F

F

F

T

F

T

F

F

F

F

T

F

T

T

F

F

F

F

F

T

F

T

F

T

V

F

F

T

T

T

F

T

V

F

T

F

T

T

T

T

F

T

F

F

T

F

T

4. Invalid.

  1. A→(B→C)
  2. ¬B …Therefore, A ∨ C
A B C ¬B A ∨ C

B→C

A→(B→C)

T

T

T

F

T

T

T

T

T

F

F

T

F

F

T

F

F

T

T

T

T

T

F

T

T

T

T

T

F

F

F

T

F

T

T

X

F

F

T

T

T

T

T

F

T

T

F

T

T

T

F

T

F

F

F

F

T

5. Valid.

  1. S→(A→J)
  2. ¬J …Therefore, A→¬ J

S

A

J

¬J

¬S

A→J

S→(A→J)

A→¬S

T

T

T

F

F

T

T

F

T

T

F

T

F

F

F

F

T

F

F

T

F

T

T

T

V

T

F

T

F

F

T

T

T

F

F

F

T

T

T

T

T

V

F

F

T

F

T

T

T

T

F

T

T

F

T

T

T

T

F

T

F

T

T

F

T

T

6. Valid.

  1. S→(A→J)
  2. J …Therefore, S→J

S

A

J

A→J

S→J

S→(A→J)

T

T

T

T

T

T

V

T

T

F

F

F

F

T

F

F

T

F

T

T

F

T

T

T

T

F

F

F

T

T

T

F

F

T

T

T

T

V

F

T

T

T

T

T

V

F

T

F

F

T

T

7. Valid.

  1. S→(A→J)
  2. A …Therefore, ¬J¬S

S

A

J

¬J

¬S

A→J

S→(A→J)

¬J→¬S

T

T

T

F

F

T

T

T

V

T

T

F

T

F

F

F

F

T

F

F

T

F

T

T

F

T

F

T

F

F

T

T

T

F

F

F

T

T

T

T

T

F

F

T

F

T

T

T

T

F

T

T

F

T

T

T

T

V

F

T

F

T

T

F

T

T

V

8. Invalid.

  1. S→(A→J)
  2. ¬J …Therefore, ¬A→¬S

S

A

J

¬J

¬A

¬S

A→J

S→(A→J)

¬A→¬S

T

T

T

F

F

F

T

T

T

T

T

F

F

T

F

F

F

T

T

F

F

T

T

F

T

T

F

X

T

F

T

F

T

F

T

T

F

F

F

F

T

T

T

T

T

T

F

F

T

F

T

T

T

T

T

F

T

T

F

F

T

T

T

T

F

T

F

T

F

T

F

T

T

9. Invalid.

  1. (E→F) ∧ (F→G)
  2. E ∨ G …Therefore, ¬F

E

F

G

¬F

E ∨ G

E→F

F→G

(E→F) ∧ (F→G)

T

T

T

F

T

T

T

T

X

T

T

F

F

T

T

F

F

T

F

F

T

T

F

T

F

T

F

T

T

T

F

T

F

F

F

F

T

F

T

T

T

F

F

T

T

T

T

T

T

F

T

T

F

T

T

T

T

F

T

F

F

F

T

F

F

10. Invalid.

  1. E→(F ∨ G)
  2. ¬F … Therefore, G→¬E

E

F

G

¬E

F ∨ G

G→¬E

¬F

(E→F) ∧ (F→G)

T

T

T

F

T

F

F

T

T

T

F

F

T

T

F

T

T

F

F

F

F

T

T

F

T

F

T

F

T

F

T

T

X

F

F

F

T

F

T

T

T

F

F

T

T

T

T

T

T

F

T

T

T

T

T

F

T

F

T

F

T

T

T

F

T

11. Valid.

  1. (E→F) ∧ (F→G)
  2. ¬F ∨ ¬G …Therefore, ¬E

E

F

G

¬E

¬F

¬G

E→F

F→G

¬F ∨ ¬G

(E→F) ∧ (F→G)

T

T

T

F

F

F

T

T

F

T

T

T

F

F

F

T

T

F

T

F

T

F

F

F

T

T

F

T

T

F

T

F

T

F

T

F

F

T

T

F

F

F

F

T

T

T

T

T

T

T

V

F

F

T

T

T

F

T

T

T

T

V

F

T

T

T

F

F

T

T

F

T

F

T

F

T

F

T

T

F

T

F

12. Invalid.

  1. (E→F) ∧ (F→G)
  2. ¬E ∨ ¬F …Therefore, ¬F ∨ ¬G

E

F

G

¬E

¬F

¬G

¬E ∨ ¬F

¬F ∨ ¬G

E→F

F→G

(E→F) ∧ (F→G)

T

T

T

F

F

F

F

F

T

T

T

T

T

F

F

F

T

F

T

T

F

F

T

F

F

F

T

T

T

T

F

T

F

T

F

T

F

T

F

T

T

F

T

F

F

F

F

T

T

T

T

T

T

T

T

F

F

T

T

T

F

T

T

T

T

T

F

T

T

T

F

F

T

F

T

T

T

X

F

T

F

T

F

T

T

T

T

F

F

13. Invalid.

  1. (W ∧ C)→(S ∧ B)
  2. ¬(C→S) …Therefore, ¬W

W

C

S

B

¬W

W ∧ C

S ∧ B

C→S

¬(C→S)

(W ∧ C)→(S ∧ B)

T

T

T

T

F

T

T

T

F

T

T

T

T

F

F

T

F

T

F

T

T

T

F

F

F

T

F

F

T

F

T

F

F

F

F

F

F

T

F

T

T

F

T

T

F

F

T

T

F

T

T

T

F

T

F

F

F

F

T

T

X

T

F

T

F

F

F

F

T

F

T

T

F

F

T

F

F

F

T

F

T

F

F

F

F

T

F

F

T

F

T

F

F

F

T

T

F

T

T

F

T

F

F

T

T

T

F

T

T

F

T

F

T

T

T

T

F

T

T

F

T

F

T

F

F

T

F

F

F

T

T

F

F

T

F

T

F

F

T

F

T

F

T

F

T

T

F

T

F

T

T

F

F

F

F

T

F

F

T

F

T

14. Valid.

  1. (W ∧ C)→(S ∨ B)
  2. ¬(C→S) …Therefore, ¬B→¬W

W

C

S

B

¬W

¬B

W ∧ C

S ∧ B

(W ∧ C)→(S ∧ B)

C→S

¬(C→S)

¬B→¬W

T

T

T

T

F

F

T

T

T

T

F

T

T

T

T

F

F

T

T

F

T

T

F

F

T

T

F

F

F

T

T

F

F

F

T

F

T

F

F

F

F

T

F

F

T

T

F

F

T

F

T

T

F

F

F

T

T

T

F

T

T

T

F

T

F

F

T

F

T

F

T

T

V

T

F

T

F

F

T

F

F

T

T

F

F

T

F

F

T

F

F

F

F

T

T

F

T

F

F

F

F

T

T

F

F

T

T

F

T

F

F

F

T

T

F

F

F

T

T

F

T

F

F

T

T

T

F

F

T

T

T

F

T

F

T

T

T

T

F

F

T

T

T

F

T

F

T

F

F

T

T

F

F

T

F

T

T

V

F

F

T

F

T

T

F

F

T

T

F

T

F

T

F

T

T

F

F

F

T

F

T

T

V

F

T

T

F

T

T

F

F

T

T

F

T

15. Invalid.

  1. (W ∧ C)→(S ∨ B)
  2. ¬(¬S→¬C) …Therefore, W ∨ B

W

C

S

B

¬S

¬C

W ∧ C

S ∧ B

(W ∧ C)→(S ∧ B)

W ∨ B

¬S→¬C

¬(¬S→¬C)

T

T

T

T

F

F

T

T

T

T

T

F

T

T

T

F

F

F

T

F

T

T

T

F

T

T

F

F

T

F

T

F

F

T

F

T

T

F

F

F

T

T

F

F

T

T

T

F

T

F

T

T

F

T

F

T

T

T

T

F

T

T

F

T

T

F

T

F

T

T

F

T

T

F

T

F

F

T

F

F

T

T

T

F

T

F

F

T

T

T

F

F

T

T

T

F

F

F

F

F

T

T

F

F

T

F

T

T

F

F

F

T

T

T

F

F

T

T

T

F

F

F

T

T

F

T

F

T

T

T

T

F

F

T

T

T

F

F

F

T

T

T

T

F

F

T

F

F

T

F

F

F

T

F

F

T

X

F

F

T

F

F

T

F

F

T

F

T

F

F

T

F

T

T

F

F

F

T

T

F

T

F

T

T

F

F

F

F

F

T

F

T

F

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