Exam 3: Inference in Propositional Logic

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. ∼(K • J) 2. I $\lor$ (L • J) / ∼K $\lor$ I

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 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?

1. ∼A ≡ B 2. A • ∼C -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. ∼(E $\lor$ F) 2) ∼F ⊃ (G ⊃ H) 3) G ≡ ∼H / ∼G

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. P ⊃ (Q ⊃ R) 2. ∼(R ⊃ S) ⊃ ∼T 3. (U $\lor$ ∼T) $\lor$ ∼(Q ⊃ S) / ∼U ⊃ ∼(P • T)

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

Mica goes swimming only if Nicole lifeguards. Pedro is free on the condition that Ona goes to the beach. Mica goes swimming unless Ona goes to the beach. Nicole doesn't lifeguard. So, either Pedro is free or Mica goes swimming. -Which of the following is the best translation into PL of this argument?

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. -Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

1. (E $\lor$ F) ⊃ (G • H) 2. ∼E ⊃ (D ⊃ ∼D) 3. ∼G $\lor$ ∼H -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)?

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). -N ? ?O 2. P ? ?Q 3. (N $\lor$ P) $\lor$ R 4. ?R / ?(O • Q)

1. P ⊃ ∼Q 2. R $\lor$ ∼∼Q 3. R ⊃ S 4. ∼S -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 any of the twenty-five rules of PL. -1. M ≡ ∼N 2. ∼M ⊃ O 3. ∼N ⊃ O / O

1. P ⊃ [Q ⊃ (P ⊃ R)] 2. P 3. Q -Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?

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. -[(J ⊃ K) ⊃ (K ⊃ L)] ⊃ (∼L ⊃ ∼J)

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

1. A ⊃ B 2. ∼B 3. ∼A ⊃ ∼C -Which of the following propositions is derivable from the given premises using the rules of section 3.1 (MP, MT, DS, HS)?

1. (F $\lor$ G) ⊃ H 2. F • E -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. -[(E ⊃ F) • (G ⊃ H)] ⊃ [(∼F $\lor$ ∼H) ⊃ (∼E $\lor$ ∼G)]

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. -[(A $\lor$ B) $\lor$ (C $\lor$ D)] ⊃ [∼(∼C ⊃ D) ⊃ (A ⊃ ∼B)]

1. ∼R ≡ S 2. T ⊃ R 3. ∼(R $\lor$ T) ⊃ U -Consider assuming 'S' 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?