Question 1

1. (A $ \lor $ B) ⊃ ∼C
2. C $ \lor $ D
3. E • A
4. F
-Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

Accepted Answer

A)  D • C 
B)  ∼F 
C)  ∼C ⊃ (A $ \lor $ 
B)  
D)  ∼E 
E)  E

Question 2

After-school activities running late entails that the buses will run late. The buses running late is a sufficient condition for either Carlos and Deandra getting home late or Edna missing dinner. Either after-school activities run late but Carlos doesn't get home late, or after-school activities run late but Deandra doesn't get home late. So, Edna misses dinner.
-Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

Accepted Answer

A)  A • (∼C $ \lor $ ∼
D)  
B)  A • (∼C • ∼
D)  
C)  A $ \lor $ (∼C $ \lor $ ∼
D)  
D)  A $ \lor $ (∼C • ∼
D)  
E)  A ⊃ [(C • 
D)  • E]

Question 3

determine whether the given proposition is a logical truth of PL or not. If it is a logical truth, provide a proof. If it is not a logical truth, provide a false valuation.
-[(K ⊃ L) ⊃ (M ⊃ N)] ⊃ [(L • M) ⊃ N]

Accepted Answer

The answer of determine whether the given proposition is a...

Question 4

determine whether the given proposition is a logical truth or not. If it is not a logical truth, select a false valuation.
-[(P $ \lor $ Q) ⊃ (R • ∼S)] ⊃ [(P $ \lor $ R) ⊃ S]

Accepted Answer

A)  Logical truth 
B)  Not a logical truth. False valuation when P, Q, R, and S are true 
C)  Not a logical truth. False valuation when P and S are true and Q and R are false 
D)  Not a logical truth. False valuation when P, Q, and R are true and S is false 
E)  Not a logical truth. False valuation when P is true and Q, R, and S are false

Question 5

After-school activities running late entails that the buses will run late. The buses running late is a sufficient condition for either Carlos and Deandra getting home late or Edna missing dinner. Either after-school activities run late but Carlos doesn't get home late, or after-school activities run late but Deandra doesn't get home late. So, Edna misses dinner.
-Working backward from the conclusion of this argument, which of the following is the most likely justification of the last step of the derivation?

Accepted Answer

A)  MP 
B)  Simp 
C)  DN 
D)  DS 
E)  HS

Question 6

determine whether the given proposition is a logical truth of PL or not. If it is a logical truth, provide a proof. If it is not a logical truth, provide a false valuation.
-[(E ⊃ F) • (G ⊃ H)] ⊃ [(∼F $ \lor $ ∼H) ⊃ (∼E $ \lor $ ∼G)]

Accepted Answer

The answer of determine whether the given proposition is a...

Question 7

derive the conclusions of each of the following arguments using the rules of inference from section 3.1 (MP, MT, DS, HS).
-1. P ? [Q ? (P ? R)]
2. P
3. Q / R

Accepted Answer

The answer of derive the conclusions of each of the...

Question 8

determine whether the given argument is valid or invalid. If it is valid, provide a derivation of the conclusion from the premises. If it is invalid, provide a counterexample.
-1. ∼(E $ \lor $ F)
2. ∼F ⊃ (G ⊃ H)
3. G ≡ ∼H / ∼G

Accepted Answer

The answer of determine whether the given argument is valid...

Question 9

1. P ⊃ [Q ⊃ (P ⊃ R)]
2. P
3. Q
-Which of the following propositions is derivable from the given premises using the rules of section 3.1 (MP, MT, DS, HS)?

Accepted Answer

A)  ∼P 
B)  R ⊃ P 
C)  R 
D)  ∼R 
E)  Q ⊃ R

Question 10

determine whether the given proposition is a logical truth or not. If it is not a logical truth, select a false valuation.
-[(K ⊃ L) ⊃ (M ⊃ N)] ⊃ [(L • M) ⊃ N]

Accepted Answer

A)  Logical truth 
B)  Not a logical truth. False valuation when K, L, and M are true and N is false 
C)  Not a logical truth. False valuation when L and M are true and K and N are false 
D)  Not a logical truth. False valuation when K, M, and N are true and L is false 
E)  Not a logical truth. False valuation when K and N are true and L and M are false

Question 11

1. ∼G • (∼H • ∼I)
2. (J $ \lor $ H) $ \lor $ K
3. J ⊃ L
4. K ⊃ M
-Which of the following propositions is derivable from the given premises using the rules available through section 3.3 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN)?

Accepted Answer

A)  L $ \lor $ M 
B)  J $ \lor $ H 
C)  G $ \lor $ I 
D)  H $ \lor $ I 
E)  H $ \lor $ K

Question 12

After-school activities running late entails that the buses will run late. The buses running late is a sufficient condition for either Carlos and Deandra getting home late or Edna missing dinner. Either after-school activities run late but Carlos doesn't get home late, or after-school activities run late but Deandra doesn't get home late. So, Edna misses dinner.
-Which of the following is the best translation into PL of this argument?

Accepted Answer

A)  A ⊃ B B ⊃ [(C • 
D)  $ \lor $ E]
(A $ \lor $ ∼
C)  $ \lor $ (A $ \lor $ ∼
D)  / E 
B)  A ⊃ B B ⊃ [(C • 
D)  $ \lor $ E]
(A • ∼
C)  $ \lor $ (A • ∼
D)  / E 
C)  A ⊃ B [(C • 
D)  $ \lor $ E] ⊃ B
(A • ∼
C)  $ \lor $ (A • ∼
D)  / E 
D)  A ⊃ B [(C • 
D)  $ \lor $ E] ⊃ B
(A $ \lor $ ∼
C)  $ \lor $ (A $ \lor $ ∼
D)  / E 
E)  A ⊃ B [(C • 
D)  $ \lor $ E] ⊃ B
(A $ \lor $ ∼
C)  • (A $ \lor $ ∼
D)  / E

Question 13

determine whether the given proposition is a logical truth of PL or not. If it is a logical truth, provide a proof. If it is not a logical truth, provide a false valuation.
-[(W $ \lor $ X) ≡ (Y $ \lor $ Z)] ⊃ [(W • ∼Y) ⊃ Z]

Accepted Answer

The answer of determine whether the given proposition is a...

Question 14

1. (Z ⊃ W) • (X ⊃ Y)
2. X $ \lor $ Z
3. ∼Y / W
-Which of the following propositions is an appropriate assumption for an indirect proof of the conclusion of the given argument?

Accepted Answer

A)  W 
B)  ∼W 
C)  Y 
D)  ∼Y 
E)  All of the above

Question 15

If a river is constantly moving and sense experience is veridical, then everything is in flux. If sense experience is veridical, but not our only means of knowledge, then not everything is in flux. So, if a river is constantly moving and sense experience is veridical, then the senses are our only means of knowledge.
-Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

Accepted Answer

A)  F ⊃ ∼(V • ∼O) 
B)  ∼∼F ⊃ ∼(V • ∼O) 
C)  ∼∼(V • ∼O) ⊃ ∼F 
D)  All of the above 
E)  Options B and C, but not option A

Question 16

1. (P $ \lor $ Q) ⊃ R
2. R ⊃ ∼Q
-Which of the following propositions is derivable from the given premises using the rules available through section 3.5 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN, Cont, Impl, Equiv, Exp, Taut)?

Accepted Answer

A)  P 
B)  ∼P 
C)  Q 
D)  ∼Q 
E)  R

Question 17

1. D • ∼E
-Which of the following propositions is an immediate (one-step) consequence in PL of the given premise?

Accepted Answer

A)  D 
B)  ∼E 
C)  ∼(D • ∼
E)  
D)  E • ∼D 
E)  None of the above

Question 18

The dentist is pleased if, and only if, I both brush and floss. The dentist is not pleased, but I brush. So, I don't floss.
-Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

Accepted Answer

A)  ∼P ≡ (B • F) 
B)  ∼P ≡ ∼(B • F) 
C)  ∼[P ≡ (B • F)] 
D)  ∼(B • F) ≡ P 
E)  None of the above

Question 19

1. ∼E ⊃ F
2. G • ∼E
-Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

Accepted Answer

A)  F ⊃ ∼E 
B)  ∼(G • ∼
E)  
C)  F 
D)  ∼E 
E)  ∼E • G

Question 20

1. (N ⊃ O) ⊃ P
2. (∼N $ \lor $ O) ≡ (∼Q ⊃ R)
3. R $ \lor $ (S ⊃ Q)
-Consider assuming '∼P' for conditional proof. Which of the following propositions is an immediate (one-step) consequence in PL of the given premises with that further assumption for conditional proof?

Accepted Answer

A)  (N ⊃ O) ≡ (∼Q ⊃ R) 
B)  (∼N ⊃ O) ≡ (∼Q ⊃ R) 
C)  (∼N $ \lor $ O) ≡ (R ⊃ ∼Q) 
D)  (∼N $ \lor $ O) ≡ (∼∼Q ⊃ ∼R) 
E)  (O ⊃ ∼N) ≡ (∼Q ⊃ R)

Exam 3: Inference in Propositional Logic

1. (A $\lor$ B) ⊃ ∼C 2. C $\lor$ D 3. E • A 4. F -Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

determine whether the given proposition is a logical truth of PL or not. If it is a logical truth, provide a proof. If it is not a logical truth, provide a false valuation. -[(K ⊃ L) ⊃ (M ⊃ N)] ⊃ [(L • M) ⊃ N]

determine whether the given proposition is a logical truth or not. If it is not a logical truth, select a false valuation. -[(P $\lor$ Q) ⊃ (R • ∼S) ] ⊃ [(P $\lor$ R) ⊃ S]

determine whether the given proposition is a logical truth of PL or not. If it is a logical truth, provide a proof. If it is not a logical truth, provide a false valuation. -[(E ⊃ F) • (G ⊃ H)] ⊃ [(∼F $\lor$ ∼H) ⊃ (∼E $\lor$ ∼G)]

derive the conclusions of each of the following arguments using the rules of inference from section 3.1 (MP, MT, DS, HS). -1. P ? [Q ? (P ? R)] 2. P 3. Q / R

determine whether the given argument is valid or invalid. If it is valid, provide a derivation of the conclusion from the premises. If it is invalid, provide a counterexample. -1. ∼(E $\lor$ F) 2. ∼F ⊃ (G ⊃ H) 3. G ≡ ∼H / ∼G

1. P ⊃ [Q ⊃ (P ⊃ R) ] 2. P 3. Q -Which of the following propositions is derivable from the given premises using the rules of section 3.1 (MP, MT, DS, HS) ?

determine whether the given proposition is a logical truth or not. If it is not a logical truth, select a false valuation. -[(K ⊃ L) ⊃ (M ⊃ N) ] ⊃ [(L • M) ⊃ N]

1. ∼G • (∼H • ∼I) 2. (J $\lor$ H) $\lor$ K 3. J ⊃ L 4. K ⊃ M -Which of the following propositions is derivable from the given premises using the rules available through section 3.3 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN) ?

determine whether the given proposition is a logical truth of PL or not. If it is a logical truth, provide a proof. If it is not a logical truth, provide a false valuation. -[(W $\lor$ X) ≡ (Y $\lor$ Z)] ⊃ [(W • ∼Y) ⊃ Z]

1. (Z ⊃ W) • (X ⊃ Y) 2. X $\lor$ Z 3. ∼Y / W -Which of the following propositions is an appropriate assumption for an indirect proof of the conclusion of the given argument?

1. (P $\lor$ Q) ⊃ R 2. R ⊃ ∼Q -Which of the following propositions is derivable from the given premises using the rules available through section 3.5 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN, Cont, Impl, Equiv, Exp, Taut) ?

1. D • ∼E -Which of the following propositions is an immediate (one-step) consequence in PL of the given premise?

The dentist is pleased if, and only if, I both brush and floss. The dentist is not pleased, but I brush. So, I don't floss. -Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

1. ∼E ⊃ F 2. G • ∼E -Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

1. (N ⊃ O) ⊃ P 2. (∼N $\lor$ O) ≡ (∼Q ⊃ R) 3. R $\lor$ (S ⊃ Q) -Consider assuming '∼P' for conditional proof. Which of the following propositions is an immediate (one-step) consequence in PL of the given premises with that further assumption for conditional proof?

Exam 3: Inference in Propositional Logic

1. (A ∨ \lor ∨ B) ⊃ ∼C 2. C ∨ \lor ∨ D 3. E • A 4. F -Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

determine whether the given proposition is a logical truth of PL or not. If it is a logical truth, provide a proof. If it is not a logical truth, provide a false valuation. -[(K ⊃ L) ⊃ (M ⊃ N)] ⊃ [(L • M) ⊃ N]

determine whether the given proposition is a logical truth or not. If it is not a logical truth, select a false valuation. -[(P ∨ \lor ∨ Q) ⊃ (R • ∼S) ] ⊃ [(P ∨ \lor ∨ R) ⊃ S]

determine whether the given proposition is a logical truth of PL or not. If it is a logical truth, provide a proof. If it is not a logical truth, provide a false valuation. -[(E ⊃ F) • (G ⊃ H)] ⊃ [(∼F ∨ \lor ∨ ∼H) ⊃ (∼E ∨ \lor ∨ ∼G)]

derive the conclusions of each of the following arguments using the rules of inference from section 3.1 (MP, MT, DS, HS). -1. P ? [Q ? (P ? R)] 2. P 3. Q / R

determine whether the given argument is valid or invalid. If it is valid, provide a derivation of the conclusion from the premises. If it is invalid, provide a counterexample. -1. ∼(E ∨ \lor ∨ F) 2. ∼F ⊃ (G ⊃ H) 3. G ≡ ∼H / ∼G

1. P ⊃ [Q ⊃ (P ⊃ R) ] 2. P 3. Q -Which of the following propositions is derivable from the given premises using the rules of section 3.1 (MP, MT, DS, HS) ?

determine whether the given proposition is a logical truth or not. If it is not a logical truth, select a false valuation. -[(K ⊃ L) ⊃ (M ⊃ N) ] ⊃ [(L • M) ⊃ N]

1. ∼G • (∼H • ∼I) 2. (J ∨ \lor ∨ H) ∨ \lor ∨ K 3. J ⊃ L 4. K ⊃ M -Which of the following propositions is derivable from the given premises using the rules available through section 3.3 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN) ?

determine whether the given proposition is a logical truth of PL or not. If it is a logical truth, provide a proof. If it is not a logical truth, provide a false valuation. -[(W ∨ \lor ∨ X) ≡ (Y ∨ \lor ∨ Z)] ⊃ [(W • ∼Y) ⊃ Z]

1. (Z ⊃ W) • (X ⊃ Y) 2. X ∨ \lor ∨ Z 3. ∼Y / W -Which of the following propositions is an appropriate assumption for an indirect proof of the conclusion of the given argument?

1. (P ∨ \lor ∨ Q) ⊃ R 2. R ⊃ ∼Q -Which of the following propositions is derivable from the given premises using the rules available through section 3.5 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN, Cont, Impl, Equiv, Exp, Taut) ?

1. D • ∼E -Which of the following propositions is an immediate (one-step) consequence in PL of the given premise?

The dentist is pleased if, and only if, I both brush and floss. The dentist is not pleased, but I brush. So, I don't floss. -Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

1. ∼E ⊃ F 2. G • ∼E -Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

1. (N ⊃ O) ⊃ P 2. (∼N ∨ \lor ∨ O) ≡ (∼Q ⊃ R) 3. R ∨ \lor ∨ (S ⊃ Q) -Consider assuming '∼P' for conditional proof. Which of the following propositions is an immediate (one-step) consequence in PL of the given premises with that further assumption for conditional proof?

1. (A $\lor$ B) ⊃ ∼C 2. C $\lor$ D 3. E • A 4. F -Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

determine whether the given proposition is a logical truth or not. If it is not a logical truth, select a false valuation. -[(P $\lor$ Q) ⊃ (R • ∼S) ] ⊃ [(P $\lor$ R) ⊃ S]

determine whether the given proposition is a logical truth of PL or not. If it is a logical truth, provide a proof. If it is not a logical truth, provide a false valuation. -[(E ⊃ F) • (G ⊃ H)] ⊃ [(∼F $\lor$ ∼H) ⊃ (∼E $\lor$ ∼G)]

determine whether the given argument is valid or invalid. If it is valid, provide a derivation of the conclusion from the premises. If it is invalid, provide a counterexample. -1. ∼(E $\lor$ F) 2. ∼F ⊃ (G ⊃ H) 3. G ≡ ∼H / ∼G

1. ∼G • (∼H • ∼I) 2. (J $\lor$ H) $\lor$ K 3. J ⊃ L 4. K ⊃ M -Which of the following propositions is derivable from the given premises using the rules available through section 3.3 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN) ?

determine whether the given proposition is a logical truth of PL or not. If it is a logical truth, provide a proof. If it is not a logical truth, provide a false valuation. -[(W $\lor$ X) ≡ (Y $\lor$ Z)] ⊃ [(W • ∼Y) ⊃ Z]

1. (Z ⊃ W) • (X ⊃ Y) 2. X $\lor$ Z 3. ∼Y / W -Which of the following propositions is an appropriate assumption for an indirect proof of the conclusion of the given argument?

1. (P $\lor$ Q) ⊃ R 2. R ⊃ ∼Q -Which of the following propositions is derivable from the given premises using the rules available through section 3.5 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN, Cont, Impl, Equiv, Exp, Taut) ?

1. (N ⊃ O) ⊃ P 2. (∼N $\lor$ O) ≡ (∼Q ⊃ R) 3. R $\lor$ (S ⊃ Q) -Consider assuming '∼P' for conditional proof. Which of the following propositions is an immediate (one-step) consequence in PL of the given premises with that further assumption for conditional proof?