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1

Solutions to Logic Tutorial 4

Q1a.

1. AB Given

2. A 1, E

3. B 1, E

4. BA 2, 3, I

Q1b.

1. AB Given

2. A 1, E

3. AB 2, I

Q1c.

1. PQ Given

2. PR Given

3. QS Given

4. P 1, E

5. Q 1, E

6. R 2, 4, E

7. S 3, 5, E

8. RS 6, 7, I

Q1d.

1. P  (QR) Given

2. PQ Assume

3. P 2, E

4. Q 2, E

5. QR 1, 3, E

6. R 4, 5, E

7. PQR 2,6, I

Q1e.

You have to show

(PQ)  (¬Q¬P) and

(¬Q¬P)  (PQ).

I will just do the first here.The second is similar.

1. (PQ) 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. (PQ) (¬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(CE)

ii) (LC)E

Showing i |- ii:

1. L(CE) Given

2. LC assume

3. L 2, E

4. CE 1, 3, E

5. C 2, E

6. E 4, 5, E

7. (LC)E 2, 6, I

Showing ii |- i:

1. (LC)E Given

2. L assume

3. C assume

4. LC 2,3, I

5. E 1, 4, E

6. CE 3, 5, I

7. L(CE) 2, 6, I

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Q3.

a. Murderer
b. Formalising the information:

1. MurdererBlackmailer

2. MurdererViolent

3. BlackmailerRich

4. RichSpendsAccount

5. Account

6. Spends

Abbreviate to:

1. MB Given

2. MV Given

3. BR Given

4. RSA Given

5. A Given

6. S Given

Deriving M:

7. B assume

8. R 3, 7, E

9. SA 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 ¬(pq)  p  ¬q

¬(pq)  ¬(¬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, ABC ├ BC

1. ABC Given

2. BC Given

3. C(BA) Given

4. ABC Given

5. A Assume

6. BC 1, 5, lemma1

7. C 2, 6, dilemma

8. (BA) 3, 7, E

9. B  ¬A 8, (a)

10. ¬A 9, E

11. ¬A 5, 10, RAA

12. BC 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. ¬( BA) 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