Exam 3: Inference in Propositional Logic

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. -Which of the following is the best translation into PL of this argument?

1. ∼(N $\lor$ O) 2. P ⊃ ∼Q -Consider assuming 'N $\lor$ P' 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?

1. (P • ∼Q) ≡ (R $\lor$ S) 2. ∼(R ≡ ∼T) 3. (T $\lor$ ∼S) • (T $\lor$ P) 4. ∼T -Which of the following propositions is derivable from the given premises using any of the twenty-five rules of PL?

1. ∼R ≡ S 2. T ⊃ R 3. ∼(R $\lor$ T) ⊃ U -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?

1. (J ≡ K) $\lor$ L 2. K ≡ L 3. ∼L -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 ⊃ Q) ⊃ R 2. ∼(S ⊃ R) / S • ∼Q

1. (J ≡ K) $\lor$ L 2. K ≡ L 3. ∼L -Which of the following propositions is derivable from the given premises 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 3.4 (MP, MT, DS, HS, Add, Conj, Simp, CD, DM, Dist, Assoc, Com, DN, Cont, Impl, Equiv, Exp, Taut). -1. (G • I) ⊃ H 2. (I ⊃ H) ⊃ F / G ⊃ F

1. P ⊃ ∼Q 2. R $\lor$ ∼∼Q 3. R ⊃ S 4. ∼S -Which of the following propositions is derivable from the given premises using the rules available through section 3.2 (MP, MT, DS, HS, Add, Conj, Simp, CD)?

1. X ⊃ (∼Y $\lor$ Z) 2. ∼X ⊃ (Y $\lor$ Z) 3. ∼Z -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?

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

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. -[F ⊃ (G $\lor$ H)] ⊃ [∼G ⊃ (F ⊃ H)]

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. A ⊃ B 2. B ⊃ (C ⊃ D) 3. A • C 4. ∼D / E

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. ∼(R $\lor$ S) 2. ∼R ⊃ (∼T ⊃ X) 3. T ≡ X / T

determine whether the argument is valid or invalid. If it is invalid, select a counterexample. -1. D ⊃ (E • F) 2) D ≡ (E $\lor$ ∼F) / ∼E ⊃ ∼D

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

1. A ⊃ B 2. B ⊃ (C ⊃ D) 3. A • C 4. ∼D -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)?

If a river is constantly moving and sense experience is veridical, then everything is in flux. If sense experience is veridical, but not our only means of knowledge, then not everything is in flux. So, if a river is constantly moving and sense experience is veridical, then the senses are our only means of knowledge. -Which of the following is the best translation into PL of this argument? A. (\bulletV)\supsetF V\cdot(\supset) /(\bullet)\supset B. (\cdot)\supset \cdot(\supset\sim) /(\bullet)\supset C. \bullet(\supset) \cdot(\supset\sim) /(\bullet)\supset D. \bullet(\supset) (\bullet\sim)\supset\sim /(\bullet)\supset E. (\bulletV)\supsetF (\bullet\sim)\supset\sim /(\bullet)\supset

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. -E ⊃ (F ⊃ E)

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. (C $\lor$ E) $\lor$ D 2. ?E / C $\lor$ D