978-1259690877 Test Bank Chapter 10 Part 2

10–21

46.

Determine which of the lettered claims below is equivalent to the following: The gun can

be sold even though it has no trigger lock. (This can be easy to do if you symbolize the

claims first and have some familiarity either with truth tables or with the Group II rules for

derivations—the truth-functional equivalences.)

A. Only if the gun has a trigger lock can it be sold.

B. The gun has no trigger lock, but it can be sold anyway.

C. If the gun cannot be sold, then it has no trigger lock.

D. If the gun has no trigger lock, then it can be sold.

Equivalent to B

47.

Use the short truth-table method to determine whether the following is valid or invalid:

P → Q

Q → R

R → S

P/∴ S v T

Valid

48.

Use the short truth-table method to determine whether the following is valid or invalid:

A → B

C → D

B v D/∴ A v C

Invalid: A = F

C = F

Make either B or D = T

49.

Use the short truth-table method to determine whether the following is valid or invalid:

A → B

~D → ~C

~D

~C → ~B/∴ ~A

Valid.

50.

Use the short truth-table method to determine whether the following is valid or invalid:

A → B

~C v D

E → F

G → H

A v C

E v G/∴ (B v D) v (F v H)

Valid.

51.

Use the short truth-table method to determine whether the following is valid or invalid:

A → B

C → D

~(~A & ~C)/∴ ~B → (D & ~B)

Valid.

52.

Use the short truth-table method to determine whether the following is valid or invalid:

~E v ~D

~(A → C)

~(~B v ~E)

~(A & B) v (C → D)/∴ ~A

Invalid: A = T

B = T

C = F

D = F

E = T

53.

Use the short truth-table method to determine whether the following is valid or invalid:

P → Q

~Q v R

~(R & ~S)

~P/∴ ~S

54.

Use the short truth-table method to determine whether the following is valid or invalid:

P → (Q & R)

R → (T v S)

~T v (U & ~W)

P/∴ ~W

Invalid: P = T

Q = T

R = T

S = T

T = F

U = T or F

W = T

55.

Use the short truth-table method to determine whether the following is valid or invalid:

P → Q

~(Q & ~R)

~R v S

~(S & ~T)

T/∴ P

Invalid: P = F OR P = F OR P = F

Q = F Q = T Q = F

R = F R = T R = T or F

S = F S = T S = T

T = T T = T T = T

56.

Use the short truth-table method to determine whether the following is valid or invalid:

M → (N → O)

(~N v O) → ~P

P/∴ ~M

57.

Using only rules from Group I, provide a deductive proof demonstrating the validity of the

following:

1. P → (Q & R) (Premise)

2. (Q & R) → S (Premise)/∴ P → S

3. P

→ S 1,2, CA

58.

Using only rules from Group I, provide a deductive proof demonstrating the validity of the

following:

1. P → Q (Premise)

2. P v R (Premise)

3. R → (S & T) (Premise)/∴ Q v (S & T)

4. Q v (S & T) 1,3,2, CD

59.

Using only rules from Group I, provide a deductive proof demonstrating the validity of the

following:

1. (P & Q) → (R v S) (Premise)

2. (R v S) → T (Premise)

3. P & Q (Premise)/∴ T

4. R v S 1,3, MP

5. T 2,4, MP

60.

Using only rules from Group I, provide a deductive proof demonstrating the validity of the

following:

1. (R v Q) & S (Premise)

2. (R v Q) → ~P (Premise)

3. P v T (Premise)/∴ T

4. R v Q 1, SIM

5.

~P 2,4, MP

6. T 3,5, DA

61.

Using only rules from Group I, provide a deductive proof demonstrating the validity of the

following:

1. P → (S v T) (Premise)

2. S → Q (Premise)

3. P (Premise)

4. T → R (Premise)/∴ Q v R

5. S v T 1,3, MP

6.

Q v R 2,4,5, CD

62.

Using only rules from Group I, provide a deductive proof demonstrating the validity of the

following:

1. ~P & Q (Premise)

2. R → P (Premise)/ ∴ Q & ~R

3.

~P 1, SIM

4.

~R 2,3, MT

5. Q 1, SIM

6. Q &

~R 4,5, CONJ

63.

Using only rules from Group I, provide a deductive proof demonstrating the validity of the

following:

1. P → M (Premise)

2. ~M v ~Q (Premise)

3. S → Q (Premise)

4. (~P v ~S) → R (Premise)/∴ R

5. ~P v ~S 1,3,2, DD

6. R 4,5, MP

64.

Using only rules from Group I, provide a deductive proof demonstrating the validity of the

following:

1. Q → (P → R) (Premise)

2. ~R & Q (Premise)/∴ ~P

3. Q 2, SIM

4. P

→

R 1,3, MP

5. ~R 2, SIM

6. ~P 4,5, MT

65.

Using only rules from Group I, provide a deductive proof demonstrating the validity of the

following:

1. (P v Q) → (X & Z) (Premise)

2. (X → W) & (Z → Y) (Premise)

3. P (Premise)/∴ W & Y

4. P v Q 3, ADD

5

.

X & Z 1,4, MP

6

.

X 5, SIM

7

.

X

→

W 2, SIM

8

.

W 7,6, MP

9

.

Z 5, SIM

10. Z

→

Y 2, SIM

11

.

Y 10,9, MP

12

.

W & Y 8,11, CONJ

66.

Using only rules from Group I, provide a deductive proof demonstrating the validity of the

following:

67.

Using rules from both Group I and Group II, construct a deduction to prove that the

following is valid.

1. P & Q (Premise)

2. R v ~(P & Q) (Premise)/∴ R

3.

~~(P & Q) 1, DN

4

.

R 2,3, DA

68.

Using rules from both Group I and Group II, construct a deduction to prove that the

following is valid.

1. ~(P v Q) (Premise)/∴ ~Q

2.

~P &

~Q 1, DEM

3. ~Q 2, SIM

69.

Using rules from both Group I and Group II, construct a deduction to prove that the

following is valid.

1. (P v Q) & (P v R) (Premise)

2. ~P (Premise)/∴ Q & R

70.

Using rules from both Group I and Group II, construct a deduction to prove that the

following is valid.

1. P v Q (Premise)

2. Q → R (Premise)/∴ ~P → R

3.

~P

→

Q 1, IMPL

4. ~P

→

R 2,3, CA

71.

Using rules from both Group I and Group II, construct a deduction to prove that the

following is valid.

1. P → Q (Premise)

2. R → Q (Premise)

3. P v R (Premise)/∴ Q

4. Q v Q 1,2,3 CD

5. Q 4, TAUT

72.

Using rules from both Group I and Group II, construct a deduction to prove that the

following is valid.

1. P v ~Q (Premise)

2. R → Q (Premise)/∴ R → P

3.

~Q v P 1, COM

4. Q

→

P 3, IMPL

5. R

→

P 2,4, CA

73.

Using rules from both Group I and Group II, construct a deduction to prove that the

following is valid.

1. P v (Q & R) (Premise)

2. Q → ~R (Premise)/∴ P

3.

~Q v

~R 2, IMPL

4.

~(Q & R) 3, DEM

5. P 1,4, DA

74.

Using rules from both Group I and Group II, construct a deduction to prove that the

following is valid.

1. (M v P) → Q (Premise)

2. Q → (~L v R) (Premise)

3. (L → R) → S (Premise)/∴ (M v P) → S

4. (M v P)

→

(~L v R) 1,2, CA

5. (M v P)

→

(L

→ R) 4, IMPL

6. (M v P)

→

S 3,5, CA

75.

Using rules from both Group I and Group II, construct a deduction to prove that the

following is valid.

1. P → ~(M & R) (Premise)

2. ~P → Q (Premise)/∴ ~Q → (~M v ~R)

This item shows the advantage of the deductive method for proving validity over even the

short truth-table method. Only a modest familiarity with CA and IMPL should enable you

to do this problem in your head.

76.

Using rules from both Group I and Group II, construct a deduction to prove that the

following is valid.

1. P v (Q & P) (Premise)

2. P → R (Premise)/∴ R

3.

(P v Q) & (P v P) 1, DIST

4.

P v P 3, SIM

5.

P 4, TAUT

6.

R 2,5, MP

77.

Using rules from both Group I and Group II, construct a deduction to prove that the

following is valid.

1. P → (Q → R) (Premise)/∴ Q → (P → R)

2. (P & Q)

→ R 1, EXPORT

3.

(Q & P)

→ R 2, COM

4.

Q

→ (P

→ R) 3, EXPORT

78.

Using rules from both Group I and Group II, construct a deduction to prove that the

following is valid.

1. (P → Q) & (R → S) (Premise)

2. Q → ~S (Premise)

3. ~T → (P & R) (Premise)/∴ T

4. P

→ Q 1, SIM

5.

R

→ S 1, SIM

6.

~Q v

~S 2, IMPL

7.

~P v

~R 4,5,6, DD

8.

~(P & R) 7, DEM

9. T 3,8, MT

79.

Using rules from both Group I and Group II, construct a deduction to prove that the

following is valid.

1. P → (Q & R) (Premise)

2. ~S → ~(Q v R) (Premise)/∴ P → S

80.

Using rules from both Group I and Group II, construct a deduction to prove that the

following is valid.

1. P → (Q & R) (Premise)

2. R → (S v T) (Premise)/∴ ~(~P v S) → T

3.

~P v (Q & R) 1, IMPL

4.

~P v (R & Q) 3, COM

5.

(~P v R) & (~P v Q) 4, DIST

6.

~P v R 5, SIM

7. P

→ R 6, IMPL

8. P

→ (S v T) 2,7, CA

9. P

→ (~S

→ T) 8, IMPL

10. (P & ~S)

→ T 9, EXPORT

11. (~~P & ~S)

→ T 10, DN

12. ~(~P v S)

→ T 11, DEM

81.

Using rules from both Group I and Group II, construct a deduction to prove that the

following is valid.

1. P → (Q & R) (Premise)

2. ~(S & Q) (Premise)

3. (R → S) & Q (Premise)/∴ ~P

4.

~S v

~Q 2, DEM

5.

~Q v

~S 4, COM

6.

Q

→ ~S 5, IMPL

7. Q 3, SIM

8.

~S 6,7, MP

9. R

→ S 3, SIM

10. ~R 9,8, MT

11. ~R v ~Q 10, ADD

12. ~(R & Q) 11, DEM

13. ~(Q & R) 12, COM

14. ~P 1,14, MT

82.

Determine whether the following symbolized argument is valid or invalid. If invalid, provide

a counterexample; if valid, construct a deduction.

P v (R & ~S)

Q → ~P/∴ Q → S

Invalid.

P = F

Q = T

R = T

S = F

83.

Determine whether the following symbolized argument is valid or invalid. If invalid, provide

a counterexample; if valid, construct a deduction.

P → (Q v R)

~S v ~T

(Q v R) → (S & T)/∴ ~P