Question 1

1. ∼W • ∼X
2. Y ⊃ W
3. ∼Y ⊃ Z
-Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

Accepted Answer

A)  W 
B)  ∼W 
C)  Y 
D)  Y ⊃ Z 
E)  Z ⊃ W 
A)  W 
B)  ∼W 
C)  Y 
D)  Y ⊃ Z 
E)  Z ⊃ W

Question 2

Explanations are good just in case understanding is increased. Understanding is not increased if, and only if, explanations are empirical deductions. So, explanations are good if, and only if, they are not logical deductions.
-Which of the following propositions is not derivable from the given premises using any of the twenty-five rules of PL and either the direct or conditional methods of proof?

Accepted Answer

A)  ∼D ⊃ ∼E 
B)  D ≡ ∼E 
C)  E ⊃ ∼D 
D)  ∼D ≡ E 
E)  D ⊃ ∼E 
A)  ∼D ⊃ ∼E 
B)  D ≡ ∼E 
C)  E ⊃ ∼D 
D)  ∼D ≡ E 
E)  D ⊃ ∼E

Question 3

1. (F • ∼ G) ⊃ H
2. J $ \lor $ F
3. ∼G • ∼I
4. ∼J / H $ \lor $ I
-Which of the following propositions is an appropriate assumption for an indirect proof of the conclusion of the given argument?

Accepted Answer

A)  H $ \lor $ I 
B)  H • I 
C)  ∼H $ \lor $ ∼I 
D)  ∼H • ∼I 
E)  None of the above 
A)  H $ \lor $ I 
B)  H • I 
C)  ∼H $ \lor $ ∼I 
D)  ∼H • ∼I 
E)  None of the above

Question 4

If neuroscience makes progress, then the mind is the brain. If the mind is soul-like, then rational thought is inexplicable. If neuroscience does not progress, then the mind is soul- like. If philosophy of mind is vacuous, then the mind is not the brain and rational thought is explicable. So, philosophy of mind is not vacuous.
-Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

Accepted Answer

A)  N $ \lor $ S 
B)  ∼∼N $ \lor $ S 
C)  S ⊃ ∼N 
D)  M ⊃ N 
E)  N $ \lor $ M 
A)  N $ \lor $ S 
B)  ∼∼N $ \lor $ S 
C)  S ⊃ ∼N 
D)  M ⊃ N 
E)  N $ \lor $ M

Question 5

[(J ⊃ K) • (K ⊃ L)] ⊃ (∼L ⊃ ∼J)
Which of the following propositions is a proper assumption for conditional proof to prove that the above wff is a logical truth of PL?

Accepted Answer

A)  [(J ⊃ K) • (K ⊃ L)] ⊃ (∼L ⊃ ∼J) 
B)  [(J ⊃ K) • (K ⊃ L)] ⊃ ∼L 
C)  (J ⊃ K) • (K ⊃ L) 
D)  J ⊃ K 
E)  J 
A)  [(J ⊃ K) • (K ⊃ L)] ⊃ (∼L ⊃ ∼J) 
B)  [(J ⊃ K) • (K ⊃ L)] ⊃ ∼L 
C)  (J ⊃ K) • (K ⊃ L) 
D)  J ⊃ K 
E)  J

Question 6

1. ∼G • (∼H • ∼I)
2. (J $ \lor $ H) $ \lor $ K
3. J ⊃ L
4. K ⊃ M
-Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

Accepted Answer

A)  J $ \lor $ H 
B)  J $ \lor $ K 
C)  ∼[G • (H • I)] 
D)  G $ \lor $ (H $ \lor $ I) 
E)  (∼H • ∼I) • ∼G 
A)  J $ \lor $ H 
B)  J $ \lor $ K 
C)  ∼[G • (H • I)] 
D)  G $ \lor $ (H $ \lor $ I) 
E)  (∼H • ∼I) • ∼G

Question 7

1. (U • Y) ⊃ Z
2. ∼Z
-Consider assuming 'Y' 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)  ∼(U • Y) 
B)  (Y • U) ⊃ Z 
C)  Y • ∼Z 
D)  All of the above 
E)  None of the above 
A)  ∼(U • Y) 
B)  (Y • U) ⊃ Z 
C)  Y • ∼Z 
D)  All of the above 
E)  None of the above

Question 8

1. A ⊃ (B $ \lor $ C)
2. ∼A $ \lor $ (D $ \lor $ E)
3. ∼(C $ \lor $ E)
-Consider assuming 'A' 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)  B $ \lor $ C 
B)  ∼∼A 
C)  ∼C • ∼E 
D)  All of the above 
E)  Options A and B, but not option C 
A)  B $ \lor $ C 
B)  ∼∼A 
C)  ∼C • ∼E 
D)  All of the above 
E)  Options A and B, but not option C

Question 9

translate the given paragraphs into arguments written in PL. Then, derive their conclusions using the rules of inference from section 3.4 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN, Cont, Impl, Equiv, Exp, Taut).
-If neuroscience makes progress, then the mind is the brain. If the mind is soul-like, then rational thought is inexplicable. If neuroscience does not progress, then the mind is soul-like. If philosophy of mind is vacuous, then the mind is not the brain and rational thought is explicable. So, philosophy of mind is not vacuous.

Accepted Answer

The answer of translate the given paragraphs into arguments written...

Question 10

derive the conclusions of each of the following arguments using the rules of inference from section 3.5 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN, Cont, Impl, Equiv, Exp, Taut).
-1. P ≡ Q
2. ∼Q / P ⊃ R

Accepted Answer

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

Question 11

translate the given paragraphs into arguments written in PL. Then, derive their conclusions using any of the twenty-five rules of PL and the direct, conditional, or indirect methods of proof.
-Owen will be happier if, and only if, he either practices the cello or quits music lessons. It is not that case that Owen quits music lessons only if he'll be happier. So, it is not the case that he both never sleeps and doesn't practice.

Accepted Answer

The answer of translate the given paragraphs into arguments written...

Question 12

1. ∼J ⊃ K
2. K ⊃ (L ⊃ M)
3. J ⊃ M
4. ∼M
-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)  ∼K 
B)  J 
C)  ∼M ⊃ L 
D)  ∼L 
E)  L 
A)  ∼K 
B)  J 
C)  ∼M ⊃ L 
D)  ∼L 
E)  L

Question 13

derive the conclusions of each of the following arguments using any of the twenty-five rules of PL.
-1. D • ?E / ?D ? E

Accepted Answer

\[\begin{array} { l l } 
\text { 1. } \m

Question 14

1. (G • I) ⊃ H
2. (I ⊃ H) ⊃ F
-Which of the following propositions is derivable from the given premises using the rules available through section 3.4 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN, Cont, Impl, Equiv, Exp, Taut)?

Accepted Answer

A)  G ⊃ I 
B)  G ⊃ H 
C)  I ⊃ H 
D)  I ⊃ F 
E)  G ⊃ F 
A)  G ⊃ I 
B)  G ⊃ H 
C)  I ⊃ H 
D)  I ⊃ F 
E)  G ⊃ F

Question 15

1. J ⊃ (K ⊃ L)
2. M ⊃ K
3. M ⊃ J
4. M • K
-Which of the following propositions is derivable from the given premises using the rules available through section 3.2 (MP, MT, DS, HS, Add, Conj, Simp, CD)?

Accepted Answer

A)  J 
B)  K 
C)  L 
D)  All of the above 
E)  Options A and B, but not option C 
A)  J 
B)  K 
C)  L 
D)  All of the above 
E)  Options A and B, but not option C

Question 16

determine whether the argument is valid or invalid. If it is invalid, select a counterexample.
-1. A ⊃ (B $ \lor $ ∼C)
2) A ≡ (B $ \lor $ C) / ∼B ⊃ ∼A

Accepted Answer

A)  Valid 
B)  Invalid. Counterexample when A, B, and C are true 
C)  Invalid. Counterexample when B and C are true and A is false 
D)  Invalid. Counterexample when A and C are true and B is false 
E)  Invalid. Counterexample when A is true and B and C are false 
A)  Valid 
B)  Invalid. Counterexample when A, B, and C are true 
C)  Invalid. Counterexample when B and C are true and A is false 
D)  Invalid. Counterexample when A and C are true and B is false 
E)  Invalid. Counterexample when A is true and B and C are false

Question 17

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. D ⊃ (E • F)
2. D ≡ (E $ \lor $ ∼F) / ∼E ⊃ ∼D

Accepted Answer

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

Question 18

1. Q ⊃ ∼(R $ \lor $ S)
2. ∼R ⊃ T
3. U ⊃ S
-Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL and either the direct or conditional methods of proof?

Accepted Answer

A)  Q ⊃ (T • ∼U) 
B)  Q ⊃ (R • S) 
C)  Q ⊃ (S • ∼T) 
D)  Q ⊃ (∼T • ∼U) 
E)  Q ⊃ (U • T) 
A)  Q ⊃ (T • ∼U) 
B)  Q ⊃ (R • S) 
C)  Q ⊃ (S • ∼T) 
D)  Q ⊃ (∼T • ∼U) 
E)  Q ⊃ (U • T)

Question 19

derive the conclusions of each of the following arguments using any of the twenty-five rules of PL and the direct, conditional, or indirect methods of proof.
-1. F ≡ (G • ∼H)
2. G ≡ (I $ \lor $ H)
3. I ⊃ ∼F / ∼F

Accepted Answer

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

Question 20

1. ∼(K • J)
2. I $ \lor $ (L • J) / ∼K $ \lor $ I
-Which of the following propositions is a likely last line of the indented sequence for an indirect proof of the given argument?

Accepted Answer

A)  I • ∼I 
B)  J • ∼J 
C)  L • ∼L 
D)  All three options above are equally likely. 
E)  Options A and B, but not options C 
A)  I • ∼I 
B)  J • ∼J 
C)  L • ∼L 
D)  All three options above are equally likely. 
E)  Options A and B, but not options C

1. ∼W • ∼X 2. Y ⊃ W 3. ∼Y ⊃ Z -Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

1. (F • ∼ G) ⊃ H 2. J $\lor$ F 3. ∼G • ∼I 4. ∼J / H $\lor$ I -Which of the following propositions is an appropriate assumption for an indirect proof of the conclusion of the given argument?

[(J ⊃ K) • (K ⊃ L)] ⊃ (∼L ⊃ ∼J) Which of the following propositions is a proper assumption for conditional proof to prove that the above wff is a logical truth of PL?

1. ∼G • (∼H • ∼I) 2. (J $\lor$ H) $\lor$ K 3. J ⊃ L 4. K ⊃ M -Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

1. (U • Y) ⊃ Z 2. ∼Z -Consider assuming 'Y' 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 ⊃ (B $\lor$ C) 2. ∼A $\lor$ (D $\lor$ E) 3. ∼(C $\lor$ E) -Consider assuming 'A' 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?

derive the conclusions of each of the following arguments using the rules of inference from section 3.5 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN, Cont, Impl, Equiv, Exp, Taut). -1. P ≡ Q 2. ∼Q / P ⊃ R

1. ∼J ⊃ K 2. K ⊃ (L ⊃ M) 3. J ⊃ M 4. ∼M -Which of the following propositions is derivable from the given premises using the rules of section 3.1 (MP, MT, DS, HS)?

derive the conclusions of each of the following arguments using any of the twenty-five rules of PL. -1. D • ?E / ?D ? E

1. (G • I) ⊃ H 2. (I ⊃ H) ⊃ F -Which of the following propositions is derivable from the given premises using the rules available through section 3.4 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN, Cont, Impl, Equiv, Exp, Taut)?

1. J ⊃ (K ⊃ L) 2. M ⊃ K 3. M ⊃ J 4. M • K -Which of the following propositions is derivable from the given premises using the rules available through section 3.2 (MP, MT, DS, HS, Add, Conj, Simp, CD)?

determine whether the argument is valid or invalid. If it is invalid, select a counterexample. -1. A ⊃ (B $\lor$ ∼C) 2) A ≡ (B $\lor$ C) / ∼B ⊃ ∼A

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. D ⊃ (E • F) 2. D ≡ (E $\lor$ ∼F) / ∼E ⊃ ∼D

1. Q ⊃ ∼(R $\lor$ S) 2. ∼R ⊃ T 3. U ⊃ S -Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL and either the direct or conditional methods of proof?

derive the conclusions of each of the following arguments using any of the twenty-five rules of PL and the direct, conditional, or indirect methods of proof. -1. F ≡ (G • ∼H) 2. G ≡ (I $\lor$ H) 3. I ⊃ ∼F / ∼F

1. ∼(K • J) 2. I $\lor$ (L • J) / ∼K $\lor$ I -Which of the following propositions is a likely last line of the indented sequence for an indirect proof of the given argument?

Introducing Logic

Propositional Logic: Syntax and Semantic

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Exam 3: Inference in Propositional Logic