<p>Exam-3-Proofs-2</p><p>The following proofs represent frequently recurring patterns of inference found in longer formal proofs of validity. Familiarity with them will be useful in doing longer proofs. Construct a formal proof of validity in F or F’ for each of the following arguments. 1. 1.  A /  A  B 2. A 3.   Intro 2, 1 4. B  Elim 3 5. A  B  Intro 2-4</p><p>Here’s a case where you can use  Elim in F’ (So I misspoke yesterday in class when I said it never makes any sense to use  Elim. The requirement is that you’ve introduced some assumption that leads to a contradiction. You still have to close out the subproof wherein you made the assumption, but here we have done so. The proof in F is just slightly longer since we have to assume B for a  Intro:</p><p>1.  A /  A  B 2. A 3.  B 4. A  A  Intro 2, 1 5. B  Intro 3-4 6. A  B  Intro 2-4</p><p>2. 1. C /  D  C 2. D 3. C Reit 1 4. D  C  Intro 2-3</p><p>3. 1. E  (F  G) /  F  (E  G) 2. F 3. E 4. F  G  Elim 1, 3 5. G  Elim 4, 2 6. E  G  Intro 3-5 7. F  (EG)  Intro 2-6 4. 1. H  (I  J) /  H  I 2. H 3. I  J  Elim 1, 2 4. I  Elim 3 5. H  I</p><p>5. 1. K  L /  K  (L  M) 2. K 3. L  Elim 1, 2 4. L  M  Intro 3 5. K  (L  M)  Intro 2-4</p><p>6.</p><p>In System F:</p><p>1. N  O /  (N P)  O 2. N  P 3. N  Elim 2 4. O  Elim 1, 3 5. (N  P)  O  Intro 2-4</p><p>In System F’:</p><p>1. N  O /  (N P)  O 2. N 3. P 4. O  Elim 1, 2 5. (N  P)  O  Intro 2, 3, 4</p><p>Remember that System F’ allows us to use multiple premises for various rules including  Intro.</p><p>7. 1. (Q  R)  S /  Q  S 2. Q 3. Q  R  Intro 2 4. S  Elim 1, 3 5. Q  S  Intro 2-4 8. 1. T  U 2. T  V / T  (U  V) 3. T 4. U  Elim 1, 2 5. V  Elim 2, 4 6. U  V  Intro 4, 5 7. T  (U  V)  Intro 3-6</p><p>9. </p><p>In System F: 1. W  X 2. Y  X /  (W  Y)  X 3. W  Y 4. W  Elim 3 5. X  Elim 1, 4 6. (W  Y)  X  Intro 3-5</p><p>In System F’: 1. W  X 2. Y  X /  (W  Y)  X 3. W 4. Y 5. X  Elim 1, 3 [or 2, 4] 6. (W  Y)  X  Intro 3, 4, 5</p><p>10. 1. Z  A 2. Z  A /  A 3. Z 4. A  Elim 1, 3 5. A 6. A Reit 5 7. A  Elim 2, 3-4, 5-6 Construct a formal proof of validity for each of the following arguments in system F’.</p><p>1. 1. A  B 2.  (C   A) / C  B 3. C 4.  A 5. C  A  Intro 3, 4 6.   Intro 5, 2 7. A  Intro 4-6 8. B  Elim 1, 7 9. C  B  Intro 3-8</p><p>Please note the correction to this problem (above) in line 2.</p><p>2. 1. (G  H)  I 2.  (G  H) / I  H 3.  I 4. G   H 5. I  Elim 1, 4 6.   Intro 3, 5 7. (G  H)  Intro 4-6 8. G 9. H 10. G  H  Intro 8, 9 11.   Intro 10, 2 12. H  Intro 9-11 13. G  H  Intro 8-12 14.   Intro 7, 13 15. I  Intro 3-14 16. I  H  Intro 15</p><p>3. 1. [(M  N)  O]  P 2. Q  [(O  M)  N] /  Q  P 3.  (  Q  P) 4. Q  P DeM 3 5. Q  Elim 4 6. P  Elim 4 7. (M  N)  O  Elim 2, 5 [Note: no need to site Assoc ] 8. P  Elim 1, 7 9.   Intro 6, 8 10. Q  P  Intro 3-9 4. 1. (V  W)  (X  W) 2.  (  X  V) /  W 3. V  W  Elim 1 4. X  W  Elim 1 5. X  V DeM 2 6. X 7. W  Elim 4, 6 8.  V 9. W  Elim 3, 8 10. W  Elim 5, 6-7, 8-9</p><p>5. 1. D  (E  F) 2.  (F   D)   G /  G  E 3. G 4.  (F   D) 5. G  Elim 2, 4 6.   Intro 3, 5 7. F  D  Intro 4-6 8. D  Elim 7 9. E  F  Elim 1, 8 10. F  Elim 7 11.  E 12. F  Elim 9, 11 13.   Intro 10, 12 14. E  Intro 11-13 15. G  E  Intro 3-14</p><p>6. 1. [H  (I  J)]  (K  J) 2. L  [I  (J  H)] /  (L  K)  J 3. L  K 4. L  Elim 3 5. K  Elim 3 6. I  (J  H)  Elim 2, 4 7. K  J  Elim 1, 6 [[Note: no need to site Assoc ] 8. J  Elim 7, 5 9. (L  K)  J 7. 1. (P  Q)  (P  R) 2. (R  S)  (R  P) /  Q  S 3. P  Q  Elim 1 4. P  R  Elim 1 5. R  S  Elim 2 6. R  P  Elim 2 7. R 8. S  Elim 5, 7 9. Q  S  Intro 8 10. P 11. Q  Elim 3, 10 12. Q  S  Intro 11 13. Q  S  Intro 6, 7-9, 10-12</p><p>8. 1. (X  Y)  (X  Y) 2.  (X  Y) /  (X  Y) 3. X  Y 4. X  Elim 3 5. X  Y  Intro 4 6.   Intro 2, 5 7. (X  Y)  Intro 3-6</p><p>9. 1. J  (J  K) 2. J  L /  (J  L)  J 3. J 4. L 5. J Reit 3 6. J 7. L  Elim 2, 6 8. J Reit 6 9. J  L  Intro 8, 7 10. (J  L)  J  Intro 3, 4, 5, 6-9 10. 1. (R  R)  (T  U) 2. R  (V  V) 3.  T /  V 4. T  U  Intro 3 5. (T  U) DeM 4 6. R  R 7. T  U  Elim 1, 6 8.   Intro 5, 7 9. (R  R)  Intro 6-8 10. R Idempotence  9 11. V  V  Elim 2, 10 12. V 13. V  Elim 11, 12 14.   Intro 12, 13 15. V</p>