1. Happy Birthday, Darwin!

2. Show that {(A v ~C)  ~B, [~B  (Q &~Q)], ~C & A} ⊦ is inconsistent in SD 1 (A v ~C)  ~BA 2 ~B  (Q & ~Q)A 3 ~C & AA

3. Show that {(A v ~C)  ~B, [~B  (Q &~Q)], ~C & A} ⊦ is inconsistent in SD 1 (A v ~C)  ~BA 2 ~B  (Q & ~Q)A 3 ~C & AA Q&E ~Q&E

4. Show that {(A v ~C)  ~B, [~B  (Q &~Q)], ~C & A} ⊦ is inconsistent in SD 1 (A v ~C)  ~BA 2 ~B  (Q & ~Q)A 3 ~C & AA Q & ~Q  E Q&E ~Q&E

5. Show that {(A v ~C)  ~B, [~B  (Q &~Q)], ~C & A} ⊦ is inconsistent in SD 1 (A v ~C)  ~BA 2 ~B  (Q & ~Q)A 3 ~C & AA ~B Q & ~Q  E Q&E ~Q&E

6. Show that {(A v ~C)  ~B, [~B  (Q &~Q)], ~C & A} ⊦ is inconsistent in SD 1 (A v ~C)  ~BA 2 ~B  (Q & ~Q)A 3 ~C & AA ~B  E Q & ~Q  E Q&E ~Q&E

7. Show that {(A v ~C)  ~B, [~B  (Q &~Q)], ~C & A} ⊦ is inconsistent in SD 1 (A v ~C)  ~BA 2 ~B  (Q & ~Q)A 3 ~C & AA A v ~C ~B  E Q & ~Q  E Q&E ~Q&E

8. Show that {(A v ~C)  ~B, [~B  (Q &~Q)], ~C & A} ⊦ is inconsistent in SD 1 (A v ~C)  ~BA 2 ~B  (Q & ~Q)A 3 ~C & AA A v ~CvI ~B  E Q & ~Q  E Q&E ~Q&E

9. Show that {(A v ~C)  ~B, [~B  (Q &~Q)], ~C & A} ⊦ is inconsistent in SD 1 (A v ~C)  ~BA 2 ~B  (Q & ~Q)A 3 ~C & AA 4 A3 &E A v ~CvI ~B  E Q & ~Q  E Q&E ~Q&E

10. Show that {(A v ~C)  ~B, [~B  (Q &~Q)], ~C & A} ⊦ is inconsistent in SD 1 (A v ~C)  ~BA 2 ~B  (Q & ~Q)A 3 ~C & AA 4 A3 &E 5 A v ~C4 vI 6 ~B 1, 5  E 7 Q & ~Q 6, 2  E 8 Q7 &E 9 ~Q7 &E

11. Show that [A  (B  C)]  [(A & B)  C] is a theorem in SD. [A  (B  C)]  [(A & B)  C]

12. Show that [A  (B  C)]  [(A & B)  C] is a theorem in SD. A  (B  C) A/  I (A & B)  C [A  (B  C)]  [(A & B)  C]  I

13. Show that [A  (B  C)]  [(A & B)  C] is a theorem in SD. A  (B  C) A/  I A & B A/  I C (A & B)  C  I [A  (B  C)]  [(A & B)  C]  I

14. Show that [A  (B  C)]  [(A & B)  C] is a theorem in SD. A  (B  C) A/  I A & B A/  I B  C C  E (A & B)  C  I [A  (B  C)]  [(A & B)  C]  I

15. Show that [A  (B  C)]  [(A & B)  C] is a theorem in SD. A  (B  C) A/  I A & B A/  I B  C  E C  E (A & B)  C  I [A  (B  C)]  [(A & B)  C]  I

16. Show that [A  (B  C)]  [(A & B)  C] is a theorem in SD. A  (B  C) A/  I A & B A/  I A &E B  C  E C  E (A & B)  C  I [A  (B  C)]  [(A & B)  C]  I

17. Show that [A  (B  C)]  [(A & B)  C] is a theorem in SD. A  (B  C) A/  I A & B A/  I A 2 &E B  C 1,3  E B 2 &E C  E (A & B)  C  I [A  (B  C)]  [(A & B)  C]  I

18. Show that [A  (B  C)]  [(A & B)  C] is a theorem in SD. 1 A  (B  C) A/  I 2 A & B A/  I 3 A 2 &E 4 B  C 1,3  E 5 B 2 &E 6 C 4,5  E 7 (A & B)  C 2-6  I 8 [A  (B  C)]  [(A & B)  C] 1-7  I

19. Show that A  ~B and B  ~A are equivalent in SD A  ~BA B  ~A

20. Show that A  ~B and B  ~A are equivalent in SD 1 A  ~BA 2 BA/  I ~A B  ~A  I

21. Show that A  ~B and B  ~A are equivalent in SD 1 A  ~BA 2 BA/  I 3 A A/~I ~A~I B  ~A  I

22. Show that A  ~B and B  ~A are equivalent in SD 1 A  ~BA 2 BA/  I 3 A A/~I 4 B2 R 5 ~B 1,3  E ~A~I B  ~A  I

23. Show that A  ~B and B  ~A are equivalent in SD 1 A  ~BA 2 BA/  I 3 A A/~I 4 B2 R 5 ~B 1,3  E 6 ~A3-5 ~I 7 B  ~A1-6  I

24. Show that A  ~B and B  ~A are equivalent in SD Here is the other derivation (you need both). 1 B  ~AA 2 AA/  I 3 B A/~I 4 A2 R 5 ~A 1,3  E 6 ~B3-5 ~I 7 A  ~B1-6  I

25. Show that (~A  B)  (A  ~B) is a theorem in SD. (~A  B)  (A  ~B)

26. Show that (~A  B)  (A  ~B) is a theorem in SD. 1 ~A  BA/  I A  ~B (~A  B)  (A  ~B)  I

27. Show that (~A  B)  (A  ~B) is a theorem in SD. 1 ~A  BA/  I A  ~B  I (~A  B)  (A  ~B)  I

28. Show that (~A  B)  (A  ~B) is a theorem in SD. 1 ~A  BA/  I 2 AA/  I ~ B ~BA/  I A A  ~B  I (~A  B)  (A  ~B)  I

29. Show that (~A  B)  (A  ~B) is a theorem in SD. 1 ~A  BA/  I 2 AA/  I ~ B ~I ~BA/  I A~E A  ~B  I (~A  B)  (A  ~B)  I

30. Show that (~A  B)  (A  ~B) is a theorem in SD. 1 ~A  BA/  I 2 AA/  I 3 BA/~I 4 A2R 5 ~A1, 3  E 6 ~B 3-5 ~I 7 ~BA/  I A8-10~E A  ~B2-6, 7-11  I (~A  B)  (A  ~B)1-12  I

31. Show that (~A  B)  (A  ~B) is a theorem in SD. 1 ~A  BA/  I 2 AA/  I 3 BA/~I 4 A2R 5 ~A1, 3  E 6 ~B 3-5 ~I 7 ~BA/  I 8 ~AA/~E 9. ~B7R 10. B1, 8  E 11 A8-10~E 12 A  ~B2-6, 7-11  I 13 (~A  B)  (A  ~B)1-12  I

32. Show that the following argument is valid in SD: (A v C) v B ----------------------- (A v B) v (B v C) 1 (A v B) v BA (A v B) v (B v C)

33. Show that the following argument is valid in SD: (A v C) v B ----------------------- (A v B) v (B v C) 1 (A v B) v BA (A v B) v (B v C)vE

34. Show that the following argument is valid in SD: (A v C) v B ----------------------- (A v B) v (B v C) 1 (A v C) v BA 2 A v CA/vE (A v B) v (B v C)1, BA/vE (A v B) v (B v C) (A v B) v (B v C)1, vE

35. Show that the following argument is valid in SD: (A v C) v B ----------------------- (A v B) v (B v C) 1 (A v C) v BA 2 A v CA/vE (A v B) v (B v C)1, BA/vE A v BvI (A v B) v (B v C)vI (A v B) v (B v C)1, vE

36. Show that the following argument is valid in SD: (A v C) v B ----------------------- (A v B) v (B v C) 1 (A v C) v BA 2 A v CA/vE 3 AA/vE 4 A v B3, vI 5 (A v B) v (B v C)4, vI 6 CA/vE 7 B v C 6 vI 8 (A v B) v (B v C)7 vI 9 (A v B) v (B v C)2, 3-5, 6-8 vE 10 BA/vE 11 A v B10 vI 12 (A v B) v (B v C)11 vI 13 (A v B) v (B v C)1, 2-9, 10-12 vE