(i.e., Part A–G). Total possible high score = 120 marks.PART A (15 QUESTIONS WORTH 2 MARKS EACH. TOTAL = 30 MARKS)Translate the following statements into symbolic form using upper case letters to represent affirmative ordinary language statements. Clearly indicate the simple statements you are symbolizing and the symbol you are using to symbolize them.1. Tanzania becomes a banking centre given that Zanzibar attracts foreign capital.2. The crops will fail unless it rains.3. Vancouver does not allow smoking in restaurants.4. Bats are mammals only if they nourish their young with milk.5. Schultz knows nothing if and only if Schultz gets a bribe.6. Roberto lacks wisdom.7. Neither birds nor snakes are mammals.8. If Walsh fails to win, then either Harper wins or Walsh and Harper are tied.9. If either David Beckham or Patrick Walsh attend the charity concert, then neither Britney Spears nor Christina Aguilera will attend.10. Either Jodie Foster or Patrick Walsh wear black to the Grammys but it is not the case that both do.11. Ozone depletion in the atmosphere is a sufficient condition for increased cancer rates.12. A necessary condition for decreased criminal activity is the legalization of cocaine.13. If evolutionary biology is correct, then higher life forms arose by chance, and if that is so, then it is not the case that there is any design in nature and divine providence is a myth.14. If Alvin has a bill, then he is not a platypus if he has feathers.15. Given that Murphy is a bat only if he can fly, Murphy is not a bat.PART B (2 QUESTIONS WORTH 5 MARKS EACH. TOTAL = 10 MARKS)Symbolize the following arguments, using symbolized statements and the relevant truth–functional connectives. Indicate the upper case letters you use to symbolize ordinary language simple statements. Discern whether the argument is valid or invalid.1. Erik attains Valhalla given that he is valiant. And Erik is depressed assuming that he is not valiant. Furthermore, Erik fails to attain Valhalla only if he is not depressed. Thus, Erik is depressed.2. If the Gulf War was about oil and if human life is more valuable than oil, then the Gulf War was immoral. Human life is more valuable than oil, but the Gulf War was not about oil. Therefore, the Gulf War was not immoral.PART C (3 QUESTIONS WORTH 5 MARKS EACH. TOTAL = 15 MARKS)Provide truth tables to determine whether the following symbolized statements are tautologous, self–contradictory, or contingent. After completing your truth table explain how it indicates your answer.1. (~Z • ~W) -> (Z ≡ W)2. [(G ⊃ (N ⊃ ~ G)] & [(N ≡ G) • (N v G)]3. B ⊃ [ ~ (A • B) ⊃ A]PART D (5 MARKS)Use the indirect method to determine whether the following arguments are valid or invalid. If the argument is invalid indicate truth–values of the simple statements that demonstrates the invalidity.1. 1. H v ~ S2. H ⊃ Z3. ~ S ⊃ P4. Therefore, P ≡ ZPART E (5 MARKS)Use truth table to obtain an answer to the following question. Be sure to show your work. And explain how your truth table indicates your answer.Cindy, Jane and Amanda, witnessed a bank robbery. At trial, Cindy testified that Lefty did not enter the bank, and if Howard pulled a gun, then Conrad collected the money. Jane testified that if Howard did not pull a gun, then Lefty entered the bank. Amanda testified that if Conrad collected the money, then Howard pulled a gun. Is it possible that all three witnesses told the truth? If so, what can we conclude about Lefty, Howard, and Conrad?PART F (EACH QUESTION IS WORTH 5 MARKS. TOTAL = 35 MARKS)Answer each question as directly as possible. One paragraph should be adequate to answer each question.1. Upon inspection, we immediately note that the following argument is valid. Why is this so? Could this argument ever be sound?~A v ~BA • BTherefore, C
2. Recall the Addition rule of inference (ADD). Conventionally, it seems suspicious that a disjunction can be derived where one of the disjuncts could be any statement imaginable. Explain why ADD is a correct rule of inference.3. Consider the following derivation taken from a proof of validity.n. (A • B) v C ….n+1 A n SIMPExplain why this derivation exhibits an incorrect use of the SIMP rule of inference.
4. Explain why the Conditional Proof (CP) is acceptable within our propositional logic natural deduction method.5. Explain why the Indirect Proof (IP) is acceptable within our propositional logic natural deduction method.6. Prove the following tautology using CP, IP, and any of the 18 rules of inference.[(A v B) • ~A] ⊃ B
7. Determine whether the following pair of symbolized statements are logically equivalent. Provide a truth table to support your answer.(H • K) v (K v M) (~ H • ~ K) v (~ K • ~ M)
PART G (EACH QUESTION IS WORTH 5 MARKS. TOTAL POINTS =20 MARKS)Provide a formal proof of validity of the following arguments using CP, IP, and any of the 18 rules of inference. Please note that I have changed the symbols used to represent conjunction and material conditional.1. 1. ~ A v ~ A2. A v B/∴ B
2. 1. (~J • K) ⊃ L2. ~ J/∴ ~ L ⊃ ~ K
3. 1. ~ R2. (R ⊃ S) ⊃ Q/∴ Q
4. 1. ~ (M ⊃ N)2. (M <-> K) ⊃ N3. K ⊃ M/∴ ~ (M ⊃ K)