1
Solutions to Logic Tutorial 4
Q1a.
1. AB Given
2. A 1, E
3. B 1, E
4. BA 2, 3, I
Q1b.
1. AB Given
2. A 1, E
3. AB 2, I
Q1c.
1. PQ Given
2. PR Given
3. QS Given
4. P 1, E
5. Q 1, E
6. R 2, 4, E
7. S 3, 5, E
8. RS 6, 7, I
Q1d.
1. P (QR) Given
2. PQ Assume
3. P 2, E
4. Q 2, E
5. QR 1, 3, E
6. R 4, 5, E
7. PQR 2,6, I
Q1e.
You have to show
(PQ) (¬Q¬P) and
(¬Q¬P) (PQ).
I will just do the first here.The second is similar.
1. (PQ) assume
2. ¬Q assume
3. P assume
4. Q 1,3, E
5. ¬P RAA, 3,4,2
6. ¬Q¬P I, 2,5
7. (PQ) (¬Q¬P) I,1,6
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Q2. Using L for “PM loses next vote”, C for “PM’s leadership is challenged”, E for “PM will call a
general election”:
i) L(CE)
ii) (LC)E
Showing i |- ii:
1. L(CE) Given
2. LC assume
3. L 2, E
4. CE 1, 3, E
5. C 2, E
6. E 4, 5, E
7. (LC)E 2, 6, I
Showing ii |- i:
1. (LC)E Given
2. L assume
3. C assume
4. LC 2,3, I
5. E 1, 4, E
6. CE 3, 5, I
7. L(CE) 2, 6, I
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Q3.
a. Murderer
b. Formalising the information:
1. MurdererBlackmailer
2. MurdererViolent
3. BlackmailerRich
4. RichSpendsAccount
5. Account
6. Spends
Abbreviate to:
1. MB Given
2. MV Given
3. BR Given
4. RSA Given
5. A Given
6. S Given
Deriving M:
7. B assume
8. R 3, 7, E
9. SA 4, 8, E
10. S 9, 5, E
11. B 7, 10, 6, RAA
12. M 1, 11, E
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Q4
a) Showing ¬(pq) p ¬q
¬(pq) ¬(¬p q) ¬¬p ¬q p ¬q
b)
I will use (a) and also
lemma 1 (I leave the proof of lemma 1 to you.):
A, ABC ├ BC
1. ABC Given
2. BC Given
3. C(BA) Given
4. ABC Given
5. A Assume
6. BC 1, 5, lemma1
7. C 2, 6, dilemma
8. (BA) 3, 7, E
9. B ¬A 8, (a)
10. ¬A 9, E
11. ¬A 5, 10, RAA
12. BC 4, 11, E
13. B assume
14. C 2, 13, E
15. B C 13, 14, I
16. B B C 13, 15, I
17. C assume
18. ¬( BA) 17, 3, E
19. B 18, (a), E
20. B C 17, 19, I
21. C B C 17, 20, I
22. B C 12, 16, 21, proof by cases