Exam 3: Inference in Propositional Logic

[(P ⊃ Q) ⊃ Q] ⊃ (P $\lor$ Q) -In a conditional proof of the given logical truth with no nested indirect proof, which of the following propositions appears?

1. F ⊃ (C $\lor$ D) 2. ∼[C $\lor$ (D $\lor$ E)] -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)?

derive the conclusions of each of the following arguments using any of the twenty-five rules of PL. -1. (P • ∼Q) ≡ (R $\lor$ S) 2. ∼(R ≡ ∼T) 3. (T $\lor$ ∼S) • (T $\lor$ P) 4. ∼T / Q

1. (G • I) ⊃ H 2. (I ⊃ H) ⊃ F -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 derivable from the given premises using any of the twenty-five rules of PL?

1. G ⊃ (H ⊃ I) 2. ∼(J ⊃ I) 3. ∼G ⊃ ∼J -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 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 $\lor$ R) 2. S ⊃ Q / (S • ∼P) ⊃ R

1. N ⊃ ∼O 2. P ⊃ ∼Q 3. (N $\lor$ P) $\lor$ R 4. ∼R -Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

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

translate the given paragraphs into arguments written in PL. Then, derive their conclusions using any of the twenty-five rules of PL and either the direct or conditional methods of proof. -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.

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. ∼G • (∼H • ∼I) 2. (J $\lor$ H) $\lor$ K 3. J ⊃ L 4. K ⊃ M / L $\lor$ M

1. A $\lor$ (B $\lor$ C) 2. B ⊃ D 3. C ⊃ E 4. (D $\lor$ E) ⊃ (A $\lor$ C) 5. ∼A -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 3.3 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN). -1. F ? (C $\lor$ D) 2. ?[C $\lor$ (D $\lor$ 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. ∼A $\lor$ B 2. ∼B ⊃ (C ⊃ D) 3. C ≡ ∼D / ∼C

1. A ⊃ (B $\lor$ C) 2. ∼A $\lor$ (D $\lor$ E) 3. ∼(C $\lor$ E) -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?

If Sheldon writes a paper, then using the copier is a necessary condition for Yessenia reading it. Yessenia doesn't read it only if Sheldon doesn't write it. So, if Sheldon writes a paper, then he uses the copier. -Which of the following is the best translation into PL of this argument?

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. -Consider assuming 'N • ∼P' for indirect proof of the given argument. Which of the following propositions is a likely last line of the indented sequence?

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

derive the conclusions of each of the following arguments using any of the twenty-five rules of PL. -1. ?A ? B 2. A • ?C / ?(B $\lor$ C)

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. -Consider assuming 'E' for an indirect proof of the given argument. Which of the following propositions is an immediate (one-step) consequence in PL of the given premises with that further assumption for indirect proof?