Exam 3: Inference in Propositional Logic

1. J ⊃ (K $\lor$ L) 2. J ≡ ∼K 3. (∼L $\lor$ M) • (∼L $\lor$ N) -Consider assuming 'J' 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?

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

E ⊃ (F ⊃ E) 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. ∼E ⊃ F 2. G • ∼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)?

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

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

1. Q ⊃ ∼(R $\lor$ S) 2. ∼R ⊃ T 3. U ⊃ S -Consider assuming 'Q' 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 any of the twenty-five rules of PL. -1. V ≡ (W $\lor$ ∼X) 2. ∼Y ≡ ∼V 3. (W ⊃ Y) ⊃ Z / Z

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

1. (A $\lor$ B) ⊃ ∼C 2. C $\lor$ D 3. E • A 4. F -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)?

1. P $\lor$ O 2. Q ⊃ ∼O -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)?

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

1. A ⊃ (B $\lor$ ∼C) 2. ∼(B $\lor$ D) -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 3.1 (MP, MT, DS, HS). -1. N $\lor$ (P • ∼R) 2. (P • ∼R) ⊃ Q 3. N ⊃ O 4. ∼O / Q

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. -Working backward from the conclusion of this argument, which of the following is the most likely justification of the last step of the derivation?

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. -Working backward from the conclusion of this argument, which of the following is the most likely justification of the last step of the derivation?

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 $\lor$ (B $\lor$ C) 2. B ⊃ D 3. C ⊃ E 4. (D $\lor$ E) ⊃ (A $\lor$ C) 5. ∼A / C

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. (Z ⊃ W) • (X ⊃ Y) 2. X $\lor$ Z 3. ∼Y / W

translate the given paragraphs into arguments written in PL. Then, derive their conclusions using any of the twenty-five rules of PL. -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.

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