8. For each of the following arguments, construct a proof of the conclusion from the given premises, and justify every step that is not a premise. These and all following proofs may use any of the eight basic rules of inference. a. D ? (A v C), D • ~ A / ? C b. (B ? A), (C ? B), ~ A / ? ~ C c. (A v ~ B), (~ C v B), ~ A / ? ~ C d. (A v B) ? ~ C, (C v D), A / ? D e. F ? (G • ~ H), (Z ? H), F / ? ~ Z f. (~ A • ~ B) ? C, (A ? D), (B ? D), ~ D / ? C g. (~ F v ~ G) ? (A v B), (F ? C), (B ? C), ~ C / ? A h. (A v B) ? C, (C v D) ? (E v F), A • ~ E / ? F i. (F v G) ? ~ A, A v W, F • T / ? W j. (A v B) ? T, Z ? (A v B), T ? W, ~ W / ? ~ Z * k. ~ A ? ~ B, A ? C, Z ? W, ~ C • ~ W / ? ~ B v W l. (A v B) ? (C v D), C ? E, A • ~ E / ? D v W * m. (A • B) ? ~ C, C v ~ D, A ? B, E • A / ? ~ D n. (~ A v ~ B) ? ~ G, ~ A ? (F ? G), (A ? D) • ~ D / ? ~ F * o. F ? (G ? ~ H), (F • ~ W) ? (G v T), F • ~ T, W ? T / ? ~ H * p. P ? (Q ? (R v S)), P • Q, S ? T, ~ T v ~ W, ~ ~ W / ? R q. (A ? (B ? C)), ~ B ? (F v G), (G • ~ H) ? (D ? B), (A • ~ C) v H, (~ H • ~ F) / ? ~ D r. (A v B) ? (C v D), (C ? E), (C v ~ F), (A • ~ E), (F v (D ? Z)) / ? Z
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Exercise 4-13 Prove valid using the eighteen valid argument forms. (Some of these proofs are difficult, and require between ten and twenty additional lines to complete.) (1) 1. R ⊃ (~ A ∙ T) 2. B ∨ ~ S 3. R ∨ S / ∴ A ⊃ B (2) 1. A ⊃ (~ C ⊃ ~ B) 2. C ⊃ (D ∙ E) / ∴ A ⊃ (B ⊃ D) (3) 1. A ∨ B 2. C 3. (A ∙ C) ⊃ D / ∴ D ∨ B (4) 1. C ∨ D 2. C ⊃ B 3. ~ C ⊃ ~ D / ∴ B (5) 1. P ⊃ [(Q ∙ R) ∨ S] 2. (Q ∙ R) ⊃ ~ P 3. T ⊃ ~ S / ∴ P ⊃ ~ T (6) 1. D ⊃ B 2. D ⊃ (B ⊃ W) 3. B ⊃ (W ⊃ S) / ∴ D ⊃ S (7) 1. A ⊃ B 2. A ∨ (B ∙ C) / ∴ B (8) 1. K ⊃ [(L ∨ M) ⊃ R] 2. (R ∨ S) ⊃ T / ∴ K ⊃ (M ⊃ T) (9) 1. (P ∨ R) ∙ (P ∨ Q) 2. (Q ∙ R) ⊃ (V ⊃ W) 3. ~ [(P ⊃ S) ⊃ ~ (S ⊃ W)] 4. ~ W / ∴ V ⊃ S (10) 1. A ⊃ B 2. A ∨ C 3. D ⊃ B 4. D ∨ ~ C / ∴ B (11) 1. (M ∨ N) ⊃ (M ⊃ ~ N) 2. ~ (N ⊃ P) ⊃ ~ (M ⊃ ~ N) 3. M ∨ N / ∴ M ∨ P (12) 1. (A ⊃ A) ⊃ (~ A ⊃ ~ A) / ∴ A ∨ ~ A (13) 1. A ≡ B 2. ~ (A ∙ ~ R) ⊃ (A ∙ S) / ∴ ~ (B ∙ S) ⊃ ~ (A ∙ R) (14) 1. A ⊃ B 2. C ⊃ D / ∴ (A ∨ C) ⊃ (B ∨ D) (15) 1. ~ [D ∙ ~ (E ∨ B)] 2. ~ (E ∨ F) 3. C ⊃ (E ∨ A) / ∴ ~ (~ A ∙ ~ B) ∨ ~ (C ∨ D) Summary We will soon have a complete list of rules—so that shown valid by the truth tables can be proved
Sri K.
11. Use one of the 3 methods of syllogistic logic to determine whether the following syllogism is valid (10 points): All tomatoes are fruits. Some fruits are not carrots. Therefore, no carrots are tomatoes. 12. Assume that the sentence "Some dogs are not wolves" is false. a) What inferences can you draw from the Traditional Square of Opposition? (10 points). b) What inferences can you draw from the Modern Square of Opposition? (10 points). 13. Use direct or indirect truth table method to determine whether the following argument is valid (20 points). S ∨ L / L → ~S // S ∨ ~L Bonus Questions 14. Use all rules of propositional logic to derive the conclusion of the argument below. Use conditional proof or indirect proof as needed (20 points). 1. S → (K ∧ F) 2. F → (G ∧ H) / S → H 15. Use all rules of predicate logic to derive the conclusion of the arguments below. Use conditional proof or indirect proof as needed (20 points). 1. (∀x)[(Kx ∨ Nx) → (Ex ∧ ~Rx)] 2. (∀x)[(Kx ∨ Sx) → (Rx ∨ Hx)] / (∀x)[Kx → (Ex ∧ Hx)]
Use the first four rules of inference (MP, MT, HS, DS) to derive the conclusion of the following symbolized argument.
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