The Following Proofs Represent Frequently Recurring Patterns of Inference Found in Longer
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Exam-3-Proofs-2
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
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:
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
2. 1. C / D C 2. D 3. C Reit 1 4. D C Intro 2-3
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 (EG) 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
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
6.
In System F:
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
In System F’:
1. N O / (N P) O 2. N 3. P 4. O Elim 1, 2 5. (N P) O Intro 2, 3, 4
Remember that System F’ allows us to use multiple premises for various rules including Intro.
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
9.
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
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
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’.
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
Please note the correction to this problem (above) in line 2.
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
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
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
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
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
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