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Happy Birthday, Darwin!

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

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Happy Birthday, Darwin!

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  1. Happy Birthday, Darwin!

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

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

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

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

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

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

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

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

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

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

  29. Show that (~A  B)  (A  ~B) is a theorem in SD. 1 ~A  B A/I 2A A/I ~B ~I ~B A/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  B A/I 2 A A/I 3 B A/~I 4 A 2R 5 ~A 1, 3 E 6 ~B 3-5 ~I 7 ~B A/I A 8-10~E A  ~B 2-6, 7-11I (~A  B)  (A  ~B) 1-12 I

  31. Show that (~A  B)  (A  ~B) is a theorem in SD. 1 ~A  B A/I 2 A A/I 3 B A/~I 4 A 2R 5 ~A 1, 3 E 6 ~B 3-5 ~I 7 ~B A/I 8 ~A A/~E • ~B 7R • B 1, 8 E 11 A 8-10~E 12 A  ~B 2-6, 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 B A (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 B A (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 B A 2 A v C A/vE (A v B) v (B v C) 1, B A/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 B A 2 A v C A/vE (A v B) v (B v C) 1, B A/vE A v B vI (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 B A 2 A v C A/vE 3 A A/vE 4 A v B 3, vI 5 (A v B) v (B v C) 4, vI 6 C A/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 B A/vE 11 A v B 10 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

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