Exam 3: Inference in Propositional Logic

derive the conclusions of each of the following arguments using the rules of inference from section 3.3 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN). -1. (A $\lor$ B) ? ?C 2. C $\lor$ D 3. E • A 4. F / D • F

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Question 1

Correct Answer:

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$\begin{array} { l l } \text { 1. } ( \mathrm { A } \vee \mathrm { B } ) \supset \sim \mathrm { C } & \\\text { 2. } \mathrm { C } \vee \mathrm { D } & \\\text { 3. } \mathrm { E } \cdot \mathrm { A } & \\\text { 4. F } & \text { / D } \cdot \mathrm { F } \\\text { 5. A } \cdot \mathrm { E } & \text { 3, Com } \\\text { 6. A } & 5 , \text { Simp } \\\text { 7. A } \vee B & 6 , \mathrm { Add } \\\text { 8. } \sim \mathrm { C } & 1,7 , \mathrm { MP }\end{array}$ $\begin{array} { l l } \text { 9. D } & 2,8 , \text { DS } \\\text { 10. D } \cdot \text { F } & 9,4 , \text { Conj }\end{array}$

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. -[(D ⊃ E) ⊃ F] ⊃ [D ⊃ (E ⊃ F)]

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Question 2

Correct Answer:

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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. ∼(M $\lor$ N) 2. ∼M ⊃ (O ⊃ P) 3. O ≡ P / O

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Question 3

Correct Answer:

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1. $\sim ( \mathrm { M } \vee \mathrm { N } )$
2. $\sim \mathrm { M } \supset ( \mathrm { O } \supset \mathrm { P } )$
3. $\mathrm { O } \equiv \mathrm { P }$ $\quad\quad\quad/O$ Invalid. Counterexample when M, N, O, and P are false

translate the given paragraphs into arguments written in PL. Then, derive their conclusions using the rules of inference from section 3.3 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN). -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.

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Question 4

If names are purely referential and do not have descriptive content, then 'Fido' means Fido. If names do have descriptive content, then 'Fido' means my idea of Fido. But Fido does not mean my idea of Fido. Names are purely referential. So, 'Fido' means Fido. -Which of the following is the best translation into PL of this argument?

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Question 5

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. -[(P • Q) ≡ (R $\lor$ S)] ⊃ [(P • ∼R) ⊃ S]

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Question 6

1. A ⊃ (B • C) 2. ∼D ⊃ ∼B 3. A / D -Which of the following propositions is an appropriate assumption for an indirect proof of the conclusion of the given argument?

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Question 7

provide a proof of the given logical truth using any of the twenty-five rules of PL and the direct, conditional, or indirect methods of proof. -(P $\lor$ Q) $\lor$ (∼P • ∼Q)

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Question 8

determine whether the given proposition is a logical truth or not. If it is not a logical truth, select a false valuation. -[(A $\lor$ B) $\lor$ (C $\lor$ D)] ⊃ [∼(∼C ⊃ D) ⊃ (A ⊃ ∼B)]

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Question 9

1. M ≡ ∼N 2. ∼M ⊃ O 3. ∼N ⊃ O -Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL?

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Question 10

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

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Question 11

derive the conclusions of each of the following arguments using the rules of inference from section 3.1 (MP, MT, DS, HS). -1. (P $\lor$ ?R) ? (P ? Q) 2. P $\lor$ ?R 3. P 4. Q ? ?S / ?S

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Question 12

If moral theories measure outcomes or consequences, then they do not concern duty. If trolley cases are compelling, then moral theories measure consequences. Moral theories concern duty. So, trolley cases are not compelling. -Which of the following is the best translation into PL of this argument?

(Multiple Choice)

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Question 13

1. P $\lor$ (Q ⊃ R) 2. R ⊃ (S • T) 3. ∼P -Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

(Multiple Choice)

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Question 14

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. G ? H 2. ?G ? (I ? J) 3. K ? L 4. I $\lor$ K 5. ?H / J $\lor$ L

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Question 15

Construct a derivation to prove that each of the following propositions is a logical truth of PL using any of the twenty-five rules and conditional proof. -[(G • H) ⊃ I] ⊃ [G ⊃ (H ⊃ I)]

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Question 16

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

(Multiple Choice)

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Question 17

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. (A ? B) • (C ? D) 2. E ? A 3. ?F $\lor$ E 4. ??F / B

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Question 18

Construct a derivation to prove that each of the following propositions is a logical truth of PL using any of the twenty-five rules and conditional proof. -(A ⊃ B) ⊃ {(C ⊃ D) ⊃ [(∼A ⊃ C) ⊃ (∼D ⊃ B)]}

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Question 19

I know something if, and only if, I have some causal contact with the physical world. If I know something, then there is no demon deceiver. I know something. So, I know something just in case both I have causal contact with the physical world and there is no demon deceiver. -Which of the following is the best translation into PL of this argument?

(Multiple Choice)

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Question 20

derive the conclusions of each of the following arguments using the rules of inference from section 3.3 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN). -1. (A $\lor$ B) ? ?C 2. C $\lor$ D 3. E • A 4. F / D • F

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. -[(D ⊃ E) ⊃ F] ⊃ [D ⊃ (E ⊃ F)]

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. ∼(M $\lor$ N) 2. ∼M ⊃ (O ⊃ P) 3. O ≡ P / O

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. -[(P • Q) ≡ (R $\lor$ S)] ⊃ [(P • ∼R) ⊃ S]

1. A ⊃ (B • C) 2. ∼D ⊃ ∼B 3. A / D -Which of the following propositions is an appropriate assumption for an indirect proof of the conclusion of the given argument?

provide a proof of the given logical truth using any of the twenty-five rules of PL and the direct, conditional, or indirect methods of proof. -(P $\lor$ Q) $\lor$ (∼P • ∼Q)

determine whether the given proposition is a logical truth or not. If it is not a logical truth, select a false valuation. -[(A $\lor$ B) $\lor$ (C $\lor$ D)] ⊃ [∼(∼C ⊃ D) ⊃ (A ⊃ ∼B)]

1. M ≡ ∼N 2. ∼M ⊃ O 3. ∼N ⊃ O -Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL?

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

derive the conclusions of each of the following arguments using the rules of inference from section 3.1 (MP, MT, DS, HS). -1. (P $\lor$ ?R) ? (P ? Q) 2. P $\lor$ ?R 3. P 4. Q ? ?S / ?S

If moral theories measure outcomes or consequences, then they do not concern duty. If trolley cases are compelling, then moral theories measure consequences. Moral theories concern duty. So, trolley cases are not compelling. -Which of the following is the best translation into PL of this argument?

1. P $\lor$ (Q ⊃ R) 2. R ⊃ (S • T) 3. ∼P -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 3.2 (MP, MT, DS, HS, Add, Conj, Simp, CD). -1. G ? H 2. ?G ? (I ? J) 3. K ? L 4. I $\lor$ K 5. ?H / J $\lor$ L

Construct a derivation to prove that each of the following propositions is a logical truth of PL using any of the twenty-five rules and conditional proof. -[(G • H) ⊃ I] ⊃ [G ⊃ (H ⊃ I)]

1. G ⊃ ∼(H $\lor$ I) 2. J $\lor$ G 3. K • H -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 3.2 (MP, MT, DS, HS, Add, Conj, Simp, CD). -1. (A ? B) • (C ? D) 2. E ? A 3. ?F $\lor$ E 4. ??F / B

Construct a derivation to prove that each of the following propositions is a logical truth of PL using any of the twenty-five rules and conditional proof. -(A ⊃ B) ⊃ {(C ⊃ D) ⊃ [(∼A ⊃ C) ⊃ (∼D ⊃ B)]}

Introducing Logic

Propositional Logic: Syntax and Semantic

Monadic Predicate Logic

Full First-Order Logic

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Exam 3: Inference in Propositional Logic

derive the conclusions of each of the following arguments using the rules of inference from section 3.3 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN). -1. (A ∨ \lor ∨ B) ? ?C 2. C ∨ \lor ∨ D 3. E • A 4. F / D • F

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. -[(D ⊃ E) ⊃ F] ⊃ [D ⊃ (E ⊃ F)]

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. ∼(M ∨ \lor ∨ N) 2. ∼M ⊃ (O ⊃ P) 3. O ≡ P / O

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. -[(P • Q) ≡ (R ∨ \lor ∨ S)] ⊃ [(P • ∼R) ⊃ S]

1. A ⊃ (B • C) 2. ∼D ⊃ ∼B 3. A / D -Which of the following propositions is an appropriate assumption for an indirect proof of the conclusion of the given argument?

provide a proof of the given logical truth using any of the twenty-five rules of PL and the direct, conditional, or indirect methods of proof. -(P ∨ \lor ∨ Q) ∨ \lor ∨ (∼P • ∼Q)

determine whether the given proposition is a logical truth or not. If it is not a logical truth, select a false valuation. -[(A ∨ \lor ∨ B) ∨ \lor ∨ (C ∨ \lor ∨ D)] ⊃ [∼(∼C ⊃ D) ⊃ (A ⊃ ∼B)]

1. M ≡ ∼N 2. ∼M ⊃ O 3. ∼N ⊃ O -Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL?

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

derive the conclusions of each of the following arguments using the rules of inference from section 3.1 (MP, MT, DS, HS). -1. (P ∨ \lor ∨ ?R) ? (P ? Q) 2. P ∨ \lor ∨ ?R 3. P 4. Q ? ?S / ?S

If moral theories measure outcomes or consequences, then they do not concern duty. If trolley cases are compelling, then moral theories measure consequences. Moral theories concern duty. So, trolley cases are not compelling. -Which of the following is the best translation into PL of this argument?

1. P ∨ \lor ∨ (Q ⊃ R) 2. R ⊃ (S • T) 3. ∼P -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 3.2 (MP, MT, DS, HS, Add, Conj, Simp, CD). -1. G ? H 2. ?G ? (I ? J) 3. K ? L 4. I ∨ \lor ∨ K 5. ?H / J ∨ \lor ∨ L

Construct a derivation to prove that each of the following propositions is a logical truth of PL using any of the twenty-five rules and conditional proof. -[(G • H) ⊃ I] ⊃ [G ⊃ (H ⊃ I)]

1. G ⊃ ∼(H ∨ \lor ∨ I) 2. J ∨ \lor ∨ G 3. K • H -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 3.2 (MP, MT, DS, HS, Add, Conj, Simp, CD). -1. (A ? B) • (C ? D) 2. E ? A 3. ?F ∨ \lor ∨ E 4. ??F / B

Construct a derivation to prove that each of the following propositions is a logical truth of PL using any of the twenty-five rules and conditional proof. -(A ⊃ B) ⊃ {(C ⊃ D) ⊃ [(∼A ⊃ C) ⊃ (∼D ⊃ B)]}

Introducing Logic

Propositional Logic: Syntax and Semantic

Monadic Predicate Logic

Full First-Order Logic

Filters

derive the conclusions of each of the following arguments using the rules of inference from section 3.3 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN). -1. (A $\lor$ B) ? ?C 2. C $\lor$ D 3. E • A 4. F / D • F

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. ∼(M $\lor$ N) 2. ∼M ⊃ (O ⊃ P) 3. O ≡ P / O

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. -[(P • Q) ≡ (R $\lor$ S)] ⊃ [(P • ∼R) ⊃ S]

provide a proof of the given logical truth using any of the twenty-five rules of PL and the direct, conditional, or indirect methods of proof. -(P $\lor$ Q) $\lor$ (∼P • ∼Q)

determine whether the given proposition is a logical truth or not. If it is not a logical truth, select a false valuation. -[(A $\lor$ B) $\lor$ (C $\lor$ D)] ⊃ [∼(∼C ⊃ D) ⊃ (A ⊃ ∼B)]

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

derive the conclusions of each of the following arguments using the rules of inference from section 3.1 (MP, MT, DS, HS). -1. (P $\lor$ ?R) ? (P ? Q) 2. P $\lor$ ?R 3. P 4. Q ? ?S / ?S

1. P $\lor$ (Q ⊃ R) 2. R ⊃ (S • T) 3. ∼P -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 3.2 (MP, MT, DS, HS, Add, Conj, Simp, CD). -1. G ? H 2. ?G ? (I ? J) 3. K ? L 4. I $\lor$ K 5. ?H / J $\lor$ L

1. G ⊃ ∼(H $\lor$ I) 2. J $\lor$ G 3. K • H -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 3.2 (MP, MT, DS, HS, Add, Conj, Simp, CD). -1. (A ? B) • (C ? D) 2. E ? A 3. ?F $\lor$ E 4. ??F / B