Question 1

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

Accepted Answer

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

Question 2

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 $ \lor $ Q) ⊃ R
2. R ⊃ ∼Q / ∼Q

Accepted Answer

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

Question 3

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.
-Consider assuming '∼D' 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 
B)  ∼∼U 
C)  ∼D ≡ U 
D)  D ≡ ∼U 
E)  D ≡ U 
A)  ∼U 
B)  ∼∼U 
C)  ∼D ≡ U 
D)  D ≡ ∼U 
E)  D ≡ U

Question 4

1. ∼(K • J)
2. I $ \lor $ (L • J) / ∼K $ \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)  K 
B)  ∼K 
C)  ∼(K $ \lor $ I) 
D)  ∼(∼K $ \lor $ I) 
E)  ∼∼(∼K $ \lor $ I) 
A)  K 
B)  ∼K 
C)  ∼(K $ \lor $ I) 
D)  ∼(∼K $ \lor $ I) 
E)  ∼∼(∼K $ \lor $ I)

Question 5

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

Accepted Answer

A)  ∼A 
B)  ∼∼B 
C)  (A $ \lor $ 
B)  ≡ (A $ \lor $ ∼
C)  
D)  Options A, B, and C 
E)  Options A and B, but not C 
A)  ∼A 
B)  ∼∼B 
C)  (A $ \lor $ 
B)  ≡ (A $ \lor $ ∼
C)  
D)  Options A, B, and C 
E)  Options A and B, but not C

Question 6

translate the given paragraphs into arguments written in PL. Then, derive their conclusions 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).
-It is not the case that if aesthetic values are objective, then we must treat them as such. So, aesthetic values are objective just in case it is not the case that we must treat them as such.

Accepted Answer

1. @#LAT-DLM&
2. @#LAT-DLM&, Impl
3. @#LAT-DLM&2, @#LAT-DLM&
4. @#LAT-DLM& 3, Com \[\begin

Question 7

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

Question 8

derive the conclusions of each of the following arguments using the rules of inference from section 3.2 (MP, MT, DS, HS, Add, Conj, Simp, CD).
-1. P ? ?Q
2. ?R $ \lor $ ??Q
3. ?R ? S
4. ?S / (?P • ??Q) • ??R

Accepted Answer

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

Question 9

1. P ≡ Q
2. ∼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 ⊃∼Q 
B)  ∼Q ⊃ ∼P 
C)  P ⊃ R 
D)  All of the above 
E)  Options A and B, but not option C 
A)  ∼P ⊃∼Q 
B)  ∼Q ⊃ ∼P 
C)  P ⊃ R 
D)  All of the above 
E)  Options A and B, but not option C

Question 10

It is not the case that if aesthetic values are objective, then we must treat them as such. So, aesthetic values are objective just in case it is not the case that we must treat them as such.
-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)  Impl 
B)  DM 
C)  DS 
D)  Equiv 
E)  Cont 
A)  Impl 
B)  DM 
C)  DS 
D)  Equiv 
E)  Cont

Question 11

1. D • ∼E
-Which of the following propositions is derivable from the given premise using any of the twenty-five rules of PL?

Accepted Answer

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

Question 12

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. M ⊃ J
2. M ⊃ (J ⊃ K)
3. J ⊃ (K ⊃ L) / M ⊃ L

Accepted Answer

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

Question 13

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

Accepted Answer

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

Question 14

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

Accepted Answer

A)  ∼(Y ≡ V) 
B)  ∼(∼Y ≡ ∼V) 
C)  Y ≡ ∼V 
D)  V ≡ ∼Y 
E)  Y ≡ V 
A)  ∼(Y ≡ V) 
B)  ∼(∼Y ≡ ∼V) 
C)  Y ≡ ∼V 
D)  V ≡ ∼Y 
E)  Y ≡ V

Question 15

If matter is atomic, then we can observe only modes. If we do not know of the world by pure reason, then, again, we can observe only modes. Either matter is atomic or we do not know of the world by pure reason. So, we can observe only modes.
-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)  MT 
B)  HS 
C)  Impl 
D)  DS 
E)  Taut 
A)  MT 
B)  HS 
C)  Impl 
D)  DS 
E)  Taut

Question 16

determine whether the argument is valid or invalid. If it is invalid, select a counterexample.
-1. I • (J $ \lor $ H)
2) I ⊃ ∼J
3) H ≡ (K $ \lor $ I) / K

Accepted Answer

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

Question 17

Either arithmetic and logic are both necessary or logic and geometry are both necessary. If all knowledge is experiential, then even logic is not necessary. So, not all knowledge is experiential.
-Which of the following is the best translation into PL of this argument?

Accepted Answer

A)  (A • L) $ \lor $ (L • G) E ⊃ ∼L / ∼E 
B)  N $ \lor $ ∼N E ⊃ L / ∼E 
C)  (A $ \lor $ L) • (L $ \lor $ G) E ⊃ ∼L / ∼E 
D)  N $ \lor $ ∼N E ⊃ ∼L / ∼E 
E)  (A • L) $ \lor $ (L • G) E ⊃ L / ∼E 
A)  (A • L) $ \lor $ (L • G) E ⊃ ∼L / ∼E 
B)  N $ \lor $ ∼N E ⊃ L / ∼E 
C)  (A $ \lor $ L) • (L $ \lor $ G) E ⊃ ∼L / ∼E 
D)  N $ \lor $ ∼N E ⊃ ∼L / ∼E 
E)  (A • L) $ \lor $ (L • G) E ⊃ L / ∼E

Question 18

(P $ \lor $ Q) $ \lor $ (∼P • ∼Q)
-Which of the following propositions is an appropriate assumption for an indirect proof of the given logical truth?

Accepted Answer

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

Question 19

1. (P $ \lor $ ∼R) ⊃ (P ⊃ Q)
2. P $ \lor $ ∼R
3. P
4. Q ⊃ ∼S
-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)  ∼R 
B)  P ⊃ R 
C)  ∼Q 
D)  ∼S 
E)  ∼P 
A)  ∼R 
B)  P ⊃ R 
C)  ∼Q 
D)  ∼S 
E)  ∼P

Question 20

[(G • H) ⊃ I] ⊃ [G ⊃ (H ⊃ I)]
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)  [(G • H) ⊃ I] ⊃ [G ⊃ (H ⊃ I)] 
B)  [(G • H) ⊃ I] ⊃ (G ⊃ H) 
C)  [(G • H) ⊃ I] ⊃ G 
D)  (G • H) ⊃ I 
E)  G • H 
A)  [(G • H) ⊃ I] ⊃ [G ⊃ (H ⊃ I)] 
B)  [(G • H) ⊃ I] ⊃ (G ⊃ H) 
C)  [(G • H) ⊃ I] ⊃ G 
D)  (G • H) ⊃ I 
E)  G • H

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

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 $\lor$ Q) ⊃ R 2. R ⊃ ∼Q / ∼Q

1. ∼(K • J) 2. I $\lor$ (L • J) / ∼K $\lor$ I -Which of the following propositions is an appropriate assumption for an indirect proof of the conclusion of the given argument?

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

1. A ⊃ (B $\lor$ ∼C) 2. D ⊃ (∼B • ∼E) -Consider assuming 'A • D' 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 3.2 (MP, MT, DS, HS, Add, Conj, Simp, CD). -1. P ? ?Q 2. ?R $\lor$ ??Q 3. ?R ? S 4. ?S / (?P • ??Q) • ??R

1. P ≡ Q 2. ∼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 derivable from the given premise using any of the twenty-five rules of PL?

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. M ⊃ J 2. M ⊃ (J ⊃ K) 3. J ⊃ (K ⊃ L) / M ⊃ L

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

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

determine whether the argument is valid or invalid. If it is invalid, select a counterexample. -1. I • (J $\lor$ H) 2) I ⊃ ∼J 3) H ≡ (K $\lor$ I) / K

Either arithmetic and logic are both necessary or logic and geometry are both necessary. If all knowledge is experiential, then even logic is not necessary. So, not all knowledge is experiential. -Which of the following is the best translation into PL of this argument?

(P $\lor$ Q) $\lor$ (∼P • ∼Q) -Which of the following propositions is an appropriate assumption for an indirect proof of the given logical truth?

1. (P $\lor$ ∼R) ⊃ (P ⊃ Q) 2. P $\lor$ ∼R 3. P 4. Q ⊃ ∼S -Which of the following propositions is derivable from the given premises using the rules of section 3.1 (MP, MT, DS, HS)?

[(G • H) ⊃ I] ⊃ [G ⊃ (H ⊃ I)] Which of the following propositions is a proper assumption for conditional proof to prove that the above wff is a logical truth of PL?

Introducing Logic

Propositional Logic: Syntax and Semantic

Monadic Predicate Logic

Full First-Order Logic

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Exam 3: Inference in Propositional Logic

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

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 ∨ \lor ∨ Q) ⊃ R 2. R ⊃ ∼Q / ∼Q

1. ∼(K • J) 2. I ∨ \lor ∨ (L • J) / ∼K ∨ \lor ∨ I -Which of the following propositions is an appropriate assumption for an indirect proof of the conclusion of the given argument?

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

1. A ⊃ (B ∨ \lor ∨ ∼C) 2. D ⊃ (∼B • ∼E) -Consider assuming 'A • D' 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 3.2 (MP, MT, DS, HS, Add, Conj, Simp, CD). -1. P ? ?Q 2. ?R ∨ \lor ∨ ??Q 3. ?R ? S 4. ?S / (?P • ??Q) • ??R