1. Introductory Logic PHI 120
Presentation: "n ->I(m) and m,n RAA(k)“
Introductory Logic PHI 120

2. TAs may collect this assignment
Homework
Get Proofs handout (online)
Identify and Solve first two ->I problems on handout.
Solve S14* : ~P -> Q, ~Q ⊢ P
Read pp.28-9 "double turnstile“
Study this presentation at home
esp. S14
All 10 rules committed to memory!!!
TAs may collect this assignment

3. The 10 Primitive Rules You should have the following in hand:
“The Rules” Handout
See bottom of handout

4. Two Rules of Importance
Arrow – Introduction: ->I
n ->I(m)
Reductio ad absurdum: RAA
m, n RAA(k)
Discharging assumption
One premise rule
Two premise rule

5. Two Rules of Importance
Arrow – Introduction: ->I
Reductio ad absurdum: RAA
Discharging assumption
n ->I(m)
m, n RAA(k)
Strategy

6. Arrow - Introduction
n ->I(m)

7. n ->I(m) S16: P -> Q, Q -> R ⊢ P -> R (1) P -> Q A
(2) Q -> R A
(3) ???

8. n ->I(m) S16: P -> Q, Q -> R ⊢ P -> R (1) P -> Q A
(2) Q -> R A
(3) ???
“P -> R” is not in the premises. Hence, we have to make it.
Strategy to make an ->I:
1) Assume the antecedent of conclusion
2) Solve for the consequent (i.e., as a conclusion)
3) Apply ->I rule to generate the conditional

9. ⊢ P -> R ⊢ P -> R ⊢ P -> R ⊢ P -> R n ->I(m)
Strategy to make an ->I:
1) Assume the antecedent of conclusion
2) Solve for the consequent (i.e., as a conclusion)
3) Apply ->I rule to generate the conditional

10. n ->I(m) S16: P -> Q, Q -> R ⊢ P -> R (1) P -> Q A
(2) Q -> R A
(3) ???
Strategy to make an ->I:
1) Assume the antecedent of conclusion
2) Solve for the consequent (i.e., as a conclusion)
3) Apply ->I rule to generate the conditional

11. n ->I(m) S16: P -> Q, Q -> R ⊢ P -> R (1) P -> Q A
(2) Q -> R A
(3) ???
possible premise of an ->E
Strategy to make an ->I:
1) Assume the antecedent of conclusion
2) Solve for the consequent (i.e., as a conclusion)
3) Apply ->I rule to generate the conditional

12. Apply ->I Strategy n ->I(m)
S16: P -> Q, Q -> R ⊢ P -> R
(1) P -> Q A
(2) Q -> R A
3 (3) P A (step 1 in strategy of ->I)
Strategy to make an ->I:
1) Assume the antecedent of conclusion
2) Solve for the consequent (i.e., as a conclusion)
3) Apply ->I rule to generate the conditional

13. n ->I(m) S16: P -> Q, Q -> R ⊢ P -> R (1) P -> Q A
(2) Q -> R A
3 (3) P A (step 1 in strategy of ->I)
Strategy to make an ->I:
1) Assume the antecedent of conclusion
2) Solve for the consequent (i.e., as a conclusion)
3) Apply ->I rule to generate the conditional

14. n ->I(m) S16: P -> Q, Q -> R ⊢ P -> R (1) P -> Q A
Read the problem properly!
S16: P -> Q, Q -> R ⊢ P -> R
(1) P -> Q A
(2) Q -> R A
3 (3) P A
(4)
What kind of statement is “R” (the consequent)?
Where is it located in premises?
Step 2 (often more than one line)
Strategy to make an ->I:
1) Assume the antecedent of conclusion
2) Solve for the consequent (i.e., as a conclusion)
3) Apply ->I rule to generate the conditional

15. n ->I(m) S16: P -> Q, Q -> R ⊢ P -> R (1) P -> Q A
(2) Q -> R A
3 (3) P A antecedent of (1)
(4)
Strategy to make an ->I:
1) Assume the antecedent of conclusion
2) Solve for the consequent (i.e., as a conclusion)
3) Apply ->I rule to generate the conditional

16. n ->I(m) S16: P -> Q, Q -> R ⊢ P -> R (1) P -> Q A
(2) Q -> R A
(3) P A
(4) 1,3 ->E
Strategy to make an ->I:
1) Assume the antecedent of conclusion
2) Solve for the consequent (i.e., as a conclusion)
3) Apply ->I rule to generate the conditional

17. n ->I(m) S16: P -> Q, Q -> R ⊢ P -> R (1) P -> Q A
(2) Q -> R A
(3) P A
(4) Q 1,3 ->E
Strategy to make an ->I:
1) Assume the antecedent of conclusion
2) Solve for the consequent (i.e., as a conclusion)
3) Apply ->I rule to generate the conditional

18. n ->I(m) S16: P -> Q, Q -> R ⊢ P -> R 1 (1) P -> Q A
(2) Q -> R A
3 (3) P A
1,3 (4) Q 1,3 ->E
Strategy to make an ->I:
1) Assume the antecedent of conclusion
2) Solve for the consequent (i.e., as a conclusion)
3) Apply ->I rule to generate the conditional

19. n ->I(m) S16: P -> Q, Q -> R ⊢ P -> R (1) P -> Q A
(2) Q -> R A
(3) P A
1,3 (4) Q 1,3 ->E antecedent of (2)
(5)
Strategy to make an ->I:
1) Assume the antecedent of conclusion
2) Solve for the consequent (i.e., as a conclusion)
3) Apply ->I rule to generate the conditional

20. n ->I(m) S16: P -> Q, Q -> R ⊢ P -> R (1) P -> Q A
(2) Q -> R A
(3) P A
1,3 (4) Q 1,3 ->E
(5) 2,4 ->E
Strategy to make an ->I:
1) Assume the antecedent of conclusion
2) Solve for the consequent (i.e., as a conclusion)
3) Apply ->I rule to generate the conditional

21. n ->I(m) S16: P -> Q, Q -> R ⊢ P -> R (1) P -> Q A
(2) Q -> R A
(3) P A
1,3 (4) Q 1,3 ->E
(5) R 2,4 ->E
Strategy to make an ->I:
1) Assume the antecedent of conclusion
2) Solve for the consequent (i.e., as a conclusion)
3) Apply ->I rule to generate the conditional

22. n ->I(m)
S16: P -> Q, Q -> R ⊢ P -> R 1 (1) P -> Q A 2 (2) Q -> R A 3 (3) P A 1,3 (4) Q 1,3 ->E 1,2,3 (5) R 2,4 ->E
Strategy to make an ->I:
1) Assume the antecedent of conclusion
2) Solve for the consequent (i.e., as a conclusion)
3) Apply ->I rule to generate the conditional

23. n ->I(m) S16: P -> Q, Q -> R ⊢ P -> R (1) P -> Q A
(2) Q -> R A
(3) P A
1,3 (4) Q 1,3 ->E
1,2,3 (5) R 2,4 ->E
(6)
Strategy to make an ->I:
1) Assume the antecedent of conclusion
2) Solve for the consequent (i.e., as a conclusion)
3) Apply ->I rule to generate the conditional

24. n ->I(m) S16: P -> Q, Q -> R ⊢ P -> R (1) P -> Q A
(2) Q -> R A
(3) P A
1,3 (4) Q 1,3 ->E
1,2,3 (5) R 2,4 ->E
(6) P -> R n ->I(m)
Step 3

25. n ->I(m) S16: P -> Q, Q -> R ⊢ P -> R (1) P -> Q A
(2) Q -> R A
(3) P A
1,3 (4) Q 1,3 ->E
1,2,3 (5) R 2,4 ->E
(6) P -> R 5 ->I(3)

26. n ->I(m) S16: P -> Q, Q -> R ⊢ P -> R (1) P -> Q A
(2) Q -> R A
(3) P A
1,3 (4) Q 1,3 ->E
1,2,3 (5) R 2,4 ->E
(6) P -> R 5 ->I(3)

27. n ->I(m) S16: P -> Q, Q -> R ⊢ P -> R (1) P -> Q A
(2) Q -> R A
(3) P A
1,3 (4) Q 1,3 ->E
1,2,3 (5) R 2,4 ->E
(6) P -> R 5 ->I(3)
This must be an assumption

28. n ->I(m)
S16: P -> Q, Q -> R ⊢ P -> R 1 (1) P -> Q A 2 (2) Q -> R A 3 (3) P A 1,3 (4) Q 1,3 ->E 1,2,3 (5) R 2,4 ->E (6) P -> R 5 ->I(3)

29. n ->I(m)
S16: P -> Q, Q -> R ⊢ P -> R 1 (1) P -> Q A 2 (2) Q -> R A 3 (3) P A 1,3 (4) Q 1,3 ->E 1,2,3 (5) R 2,4 ->E 1,2 (6) P -> R 5 ->I(3)

30. n ->I(m)
S16: P -> Q, Q -> R ⊢ P -> R 1 (1) P -> Q A 2 (2) Q -> R A 3 (3) P A 1,3 (4) Q 1,3 ->E 1,2,3 (5) R 2,4 ->E 1,2 (6) P -> R 5 ->I(3)
The Two Questions
Is (6) the conclusion of the sequent?
Are the assumptions correct?

31. n ->I(m) must be an assumption Any kind of wff
(will be the consequent)
Any kind of wff
(will be the antecedent)

32. Reductio ad absurdum
m,n RAA(k)

33. The Key to RAA Denials A B
If the proof contains incompatible premises, you are allowed to deny any assumption within the proof.
m, n RAA(k)
Premises: denials of one another
Conclusion: will be the denial of some assumption (k)

34. P & Q, ~P ⊢ ~R 1 (1) P & Q A 2 (2) ~P A (3) ?? The Basic Assumptions
If the proof contains incompatible premises, you are allowed to deny any assumption within the proof.
m, n RAA(k)
Premises: denials of one another
Conclusion: will be the denial of some assumption (k)

35. P & Q, ~P ⊢ ~R RAA 1 (1) P & Q A 2 (2) ~P A (3) ??
Elimination won’t work
Introduction won’t work
RAA
If the proof contains incompatible premises, you are allowed to deny any assumption within the proof
m, n RAA(k)
Premises: denials of one another
Conclusion: will be the denial of some assumption (k)

36. P & Q, ~P ⊢ ~R 1 (1) P & Q A 2 (2) ~P A (3) ??
Strategy of RAA:
1) Assume the denial of the conclusion
1 (1) P & Q A 2 (2) ~P A (3) ??
2) Derive a contradiction.
3) Use RAA to deny/discharge an assumption
If the proof contains incompatible premises, you are allowed to deny any assumption within the proof
m, n RAA(k)
Premises: denials of one another
Conclusion: will be the denial of some assumption (k)

37. P & Q, ~P ⊢ ~R 1 (1) P & Q A 2 (2) ~P A (3) ?? Strategy of RAA:
1) Assume the denial of the conclusion
Strategy of RAA:
1) Assume the denial of the conclusion
1 (1) P & Q A 2 (2) ~P A (3) ??
2) Derive a contradiction.
3) Use RAA to deny/discharge an assumption

38. P & Q, ~P ⊢ ~R 1 (1) P & Q A 2 (2) ~P A (3) A Strategy of RAA:
1) Assume the denial of the conclusion
Strategy of RAA:
1) Assume the denial of the conclusion
1 (1) P & Q A 2 (2) ~P A (3) A
2) Derive a contradiction.
3) Use RAA to deny/discharge an assumption

39. P & Q, ~P ⊢ ~R 1 (1) P & Q A 2 (2) ~P A (3) R A Strategy of RAA:
1) Assume the denial of the conclusion
Strategy of RAA:
1) Assume the denial of the conclusion
1 (1) P & Q A 2 (2) ~P A (3) R A
2) Derive a contradiction.
3) Use RAA to deny/discharge an assumption

40. P & Q, ~P ⊢ ~R 1 (1) P & Q A 2 (2) ~P A 3 (3) R A Strategy of RAA:
1) Assume the denial of the conclusion
Strategy of RAA:
1) Assume the denial of the conclusion
1 (1) P & Q A 2 (2) ~P A 3 (3) R A
2) Derive a contradiction.
3) Use RAA to deny/discharge an assumption

41. P & Q, ~P ⊢ ~R 1 (1) P & Q A 2 (2) ~P A 3 (3) R A Strategy of RAA:
1) Assume the denial of the conclusion
1 (1) P & Q A 2 (2) ~P A 3 (3) R A
2) Derive a contradiction.
2) Derive a contradiction.
3) Use RAA to deny/discharge an assumption

42. P & Q, ~P ⊢ ~R 1 (1) P & Q A 2 (2) ~P A 3 (3) R A (4) ???
Strategy of RAA:
1) Assume the denial of the conclusion
1 (1) P & Q A 2 (2) ~P A 3 (3) R A (4) ???
2) Derive a contradiction.
2) Derive a contradiction.
3) Use RAA to deny/discharge an assumption

43. P & Q, ~P ⊢ ~R 1 (1) P & Q A 2 (2) ~P A 3 (3) R A (4) 1 &E
Strategy of RAA:
1) Assume the denial of the conclusion
1 (1) P & Q A 2 (2) ~P A 3 (3) R A (4) 1 &E
2) Derive a contradiction.
2) Derive a contradiction.
3) Use RAA to deny/discharge an assumption

44. P & Q, ~P ⊢ ~R 1 (1) P & Q A 2 (2) ~P A 3 (3) R A (4) P 1 &E
Strategy of RAA:
1) Assume the denial of the conclusion
1 (1) P & Q A 2 (2) ~P A 3 (3) R A (4) P 1 &E
2) Derive a contradiction.
2) Derive a contradiction.
3) Use RAA to deny/discharge an assumption

45. P & Q, ~P ⊢ ~R 1 (1) P & Q A 2 (2) ~P A 3 (3) R A 1 (4) P 1 &E
Strategy of RAA:
1) Assume the denial of the conclusion
1 (1) P & Q A 2 (2) ~P A 3 (3) R A 1 (4) P 1 &E
2) Derive a contradiction.
2) Derive a contradiction.
3) Use RAA to deny/discharge an assumption

46. P & Q, ~P ⊢ ~R 1 (1) P & Q A 2 (2) ~P A 3 (3) R A 1 (4) P 1 &E
Strategy of RAA:
1) Assume the denial of the conclusion
1 (1) P & Q A 2 (2) ~P A 3 (3) R A 1 (4) P 1 &E
2) Derive a contradiction.
2) Derive a contradiction.
3) Use RAA to deny/discharge an assumption

47. P & Q, ~P ⊢ ~R 1 (1) P & Q A 2 (2) ~P A 3 (3) R A 1 (4) P 1 &E
Strategy of RAA:
1) Assume the denial of the conclusion
1 (1) P & Q A 2 (2) ~P A 3 (3) R A 1 (4) P 1 &E
2) Derive a contradiction.
2) Derive a contradiction.
3) Use RAA to deny/discharge an assumption
3) Use RAA to deny/discharge an assumption

48. P & Q, ~P ⊢ ~R
1 (1) P & Q A 2 (2) ~P A 3 (3) R A 1 (4) P 1 &E (5) m, n RAA(k)
Premises: denials of one another

49. P & Q, ~P ⊢ ~R
1 (1) P & Q A 2 (2) ~P A 3 (3) R A 1 (4) P 1 &E (5) 2, 4 RAA(k)
Conclusion: will be the denial of some assumption (k)

50. P & Q, ~P ⊢ ~R
1 (1) P & Q A 2 (2) ~P A 3 (3) R A 1 (4) P 1 &E (5) 2, 4 RAA(k)
The
Basic
Assumptions
Conclusion: will be the denial of some assumption (k)

51. P & Q, ~P ⊢ ~R
1 (1) P & Q A 2 (2) ~P A 3 (3) R A 1 (4) P 1 &E (5) 2, 4 RAA(k)
Conclusion: will be the denial of some assumption (k)

52. P & Q, ~P ⊢ ~R
1 (1) P & Q A 2 (2) ~P A 3 (3) R A 1 (4) P 1 &E (5) 2, 4 RAA(3)
Conclusion: will be the denial of some assumption (k)

53. P & Q, ~P ⊢ ~R
1 (1) P & Q A 2 (2) ~P A 3 (3) R A 1 (4) P 1 &E (5) ~R 2, 4 RAA(3)
Conclusion: will be the denial of some assumption (k)

54. Don't forget to define the assumption set!
P & Q, ~P ⊢ ~R
1 (1) P & Q A 2 (2) ~P A 3 (3) R A 1 (4) P 1 &E (5) ~R 2, 4 RAA(3)
Don't forget to define the assumption set!

55. Don't forget to define the assumption set!
P & Q, ~P ⊢ ~R
1 (1) P & Q A 2 (2) ~P A 3 (3) R A 1 (4) P 1 &E (5) ~R 2, 4 RAA(3)
Don't forget to define the assumption set!

56. Don't forget to define the assumption set!
P & Q, ~P ⊢ ~R
1 (1) P & Q A 2 (2) ~P A 3 (3) R A 1 (4) P 1 &E (5) ~R 2, 4 RAA(3)
Don't forget to define the assumption set!

57. Don't forget to define the assumption set!
P & Q, ~P ⊢ ~R
1 (1) P & Q A 2 (2) ~P A 3 (3) R A 1 (4) P 1 &E 1,2 (5) ~R 2, 4 RAA(3)
Don't forget to define the assumption set!

58. P & Q, ~P ⊢ ~R
1 (1) P & Q A 2 (2) ~P A 3 (3) R A 1 (4) P 1 &E 1,2 (5) ~R 2, 4 RAA(3)

59. n ->I(m) m,n RAA(k) must be an assumption Any kind of wff
(will be the consequent)
Any kind of wff
(will be the antecedent)
m,n RAA(k)
Premises: denials of one another
Conclusion: will be the denial of assumption: k

60. m,n RAA (k)
Solve S14 for Homework

61. TAs may collect this assignment
Homework
Get Proofs handout (online)
Identify and Solve first two ->I problems on handout.
Solve S14 : ~P -> Q, Q ⊢ P
Read pp.28-9 "double turnstile“
Study this presentation at home
esp. S14
TAs may collect this assignment

62. m,n RAA(k)
S14: ~P -> Q, ~Q ⊢ P

63. m,n RAA(k) S14: ~P -> Q, ~Q ⊢ P (1) ~P -> Q A (2) ~Q A (3)
Note: neither introduction nor elimination strategy will work for “P”
Strategy of RAA:
1) Assume the denial of the conclusion
2) Derive a contradiction
3) Use RAA to deny/discharge an assumption

64. m,n RAA(k) S14: ~P -> Q, ~Q ⊢ P (1) ~P -> Q A (2) ~Q A
(3) (first step of RAA)
Strategy of RAA:
1) Assume the denial of the conclusion
2) Derive a contradiction
3) Use RAA to deny/discharge an assumption

65. m,n RAA(k) S14: ~P -> Q, ~Q ⊢ P (1) ~P -> Q A (2) ~Q A
(3) A (assume)
Strategy of RAA:
1) Assume the denial of the conclusion
2) Derive a contradiction
3) Use RAA to deny/discharge an assumption

66. m,n RAA(k) S14: ~P -> Q, ~Q ⊢ P (1) ~P -> Q A (2) ~Q A
(3) ~P A (denial of conclusion)
Strategy of RAA:
1) Assume the denial of the conclusion
2) Derive a contradiction
3) Use RAA to deny/discharge an assumption

67. m,n RAA(k) S14: ~P -> Q, ~Q ⊢ P (1) ~P -> Q A (2) ~Q A
3 (3) ~P A (denial of conclusion)
Strategy of RAA:
1) Assume the denial of the conclusion
2) Derive a contradiction
3) Use RAA to deny/discharge an assumption

68. m,n RAA(k) S14: ~P -> Q, ~Q ⊢ P (1) ~P -> Q A (2) ~Q A (3) ~P A
Step Back. Read the premises.
Strategy of RAA:
1) Assume the denial of the conclusion
2) Derive a contradiction
3) Use RAA to deny/discharge an assumption

69. m,n RAA(k) S14: ~P -> Q, ~Q ⊢ P (1) ~P -> Q A (possible –>E)
3 (3) ~P A
Strategy of RAA:
1) Assume the denial of the conclusion
2) Derive a contradiction
3) Use RAA to deny/discharge an assumption

70. m,n RAA(k) S14: ~P -> Q, ~Q ⊢ P (1) ~P -> Q A (possible –>E)
3 (3) ~P A (antecedent)
Strategy of RAA:
1) Assume the denial of the conclusion
2) Derive a contradiction
3) Use RAA to deny/discharge an assumption

71. m,n RAA(k) S14: ~P -> Q, ~Q ⊢ P (1) ~P -> Q A (possible –>E)
(2) ~Q A (denial of consequent)
3 (3) ~P A (antecedent)
Strategy of RAA:
1) Assume the denial of the conclusion
2) Derive a contradiction
3) Use RAA to deny/discharge an assumption

72. m,n RAA(k) S14: ~P -> Q, ~Q ⊢ P (1) ~P -> Q A (2) ~Q A (3) ~P A
(4) ??
Strategy of RAA:
1) Assume the denial of the conclusion
2) Derive a contradiction
3) Use RAA to deny/discharge an assumption

73. m,n RAA(k) S14: ~P -> Q, ~Q ⊢ P (1) ~P -> Q A (2) ~Q A (3) ~P A
(4) 1, 3 ->E
Strategy of RAA:
1) Assume the denial of the conclusion
2) Derive a contradiction
3) Use RAA to deny/discharge an assumption

74. m,n RAA(k) S14: ~P -> Q, ~Q ⊢ P (1) ~P -> Q A (2) ~Q A (3) ~P A
(4) Q 1, 3 ->E
Strategy of RAA:
1) Assume the denial of the conclusion
2) Derive a contradiction
3) Use RAA to deny/discharge an assumption

75. m,n RAA(k) S14: ~P -> Q, ~Q ⊢ P 1 (1) ~P -> Q A (2) ~Q A
3 (3) ~P A
(4) Q 1, 3 ->E
Strategy of RAA:
1) Assume the denial of the conclusion
2) Derive a contradiction
3) Use RAA to deny/discharge an assumption

76. m,n RAA(k) S14: ~P -> Q, ~Q ⊢ P 1 (1) ~P -> Q A (2) ~Q A
3 (3) ~P A
1,3 (4) Q 1, 3 ->E
Strategy of RAA:
1) Assume the denial of the conclusion
2) Derive a contradiction
3) Use RAA to deny/discharge an assumption

77. m,n RAA(k) S14: ~P -> Q, ~Q ⊢ P 1 (1) ~P -> Q A (2) ~Q A
3 (3) ~P A
1,3 (4) Q 1, 3 ->E
(5) ??
Strategy of RAA:
1) Assume the denial of the conclusion
2) Derive a contradiction
3) Use RAA to deny/discharge an assumption

78. m,n RAA(k) S14: ~P -> Q, ~Q ⊢ P 1 (1) ~P -> Q A (2) ~Q A
3 (3) ~P A
1,3 (4) Q 1, 3 ->E
(5) ??
Strategy of RAA:
1) Assume the denial of the conclusion
2) Derive a contradiction
3) Use RAA to deny/discharge an assumption

79. m,n RAA(k) S14: ~P -> Q, ~Q ⊢ P 1 (1) ~P -> Q A (2) ~Q A
3 (3) ~P A
1,3 (4) Q 1, 3 ->E
(5) 2, 4 RAA(?)
Strategy of RAA:
1) Assume the denial of the conclusion
2) Derive a contradiction
3) Use RAA to deny/discharge an assumption

80. m,n RAA(k) S14: ~P -> Q, ~Q ⊢ P 1 (1) ~P -> Q A (2) ~Q A
3 (3) ~P A
1,3 (4) Q 1, 3 ->E
(5) 2, 4 RAA(?)
Question: which assumption will we discharge?
Strategy of RAA:
1) Assume the denial of the conclusion
2) Derive a contradiction
3) Use RAA to deny/discharge an assumption

81. m,n RAA(k) S14: ~P -> Q, ~Q ⊢ P 1 (1) ~P -> Q A (2) ~Q A
3 (3) ~P A
1,3 (4) Q 1, 3 ->E
(5) 2, 4 RAA(?)
Never discharge your basic premises!
Strategy of RAA:
1) Assume the denial of the conclusion
2) Derive a contradiction
3) Use RAA to deny/discharge an assumption

82. m,n RAA(k) S14: ~P -> Q, ~Q ⊢ P 1 (1) ~P -> Q A (2) ~Q A
3 (3) ~P A
1,3 (4) Q 1, 3 ->E
(5) 2, 4 RAA(3)
The sole remaining assumption
Strategy of RAA:
1) Assume the denial of the conclusion
2) Derive a contradiction
3) Use RAA to deny/discharge an assumption

83. Conclusion of RAA: denial of discharged assumption
m,n RAA(k)
S14: ~P -> Q, ~Q ⊢ P
1 (1) ~P -> Q A
(2) ~Q A
3 (3) ~P A
1,3 (4) Q 1, 3 ->E
(5) ? 2, 4 RAA(3)
Conclusion of RAA: denial of discharged assumption
Strategy of RAA:
1) Assume the denial of the conclusion
2) Derive a contradiction
3) Use RAA to deny/discharge an assumption

84. Conclusion of RAA: denial of discharged assumption
m,n RAA(k)
S14: ~P -> Q, ~Q ⊢ P
1 (1) ~P -> Q A
(2) ~Q A
3 (3) ~P A
1,3 (4) Q 1, 3 ->E
(5) P 2, 4 RAA(3)
Conclusion of RAA: denial of discharged assumption
Strategy of RAA:
1) Assume the denial of the conclusion
2) Derive a contradiction
3) Use RAA to deny/discharge an assumption

85. m,n RAA(k)
S14: ~P -> Q, ~Q ⊢ P 1 (1) ~P -> Q A 2 (2) ~Q A 3 (3) ~P A 1,3 (4) Q 1, 3 ->E (5) P 2, 4 RAA(3)
Strategy of RAA:
1) Assume the denial of the conclusion
2) Derive a contradiction
3) Use RAA to deny/discharge an assumption

86. m,n RAA(k)
S14: ~P -> Q, ~Q ⊢ P 1 (1) ~P -> Q A 2 (2) ~Q A 3 (3) ~P A 1,3 (4) Q 1, 3 ->E (5) P 2, 4 RAA(3)
Strategy of RAA:
1) Assume the denial of the conclusion
2) Derive a contradiction
3) Use RAA to deny/discharge an assumption

87. m,n RAA(k)
S14: ~P -> Q, ~Q ⊢ P 1 (1) ~P -> Q A 2 (2) ~Q A 3 (3) ~P A 1,3 (4) Q 1, 3 ->E 1,2 (5) P 2, 4 RAA(3)
Strategy of RAA:
1) Assume the denial of the conclusion
2) Derive a contradiction
3) Use RAA to deny/discharge an assumption

88. m,n RAA(k)
S14: ~P -> Q, ~Q ⊢ P 1 (1) ~P -> Q A 2 (2) ~Q A 3 (3) ~P A 1,3 (4) Q 1, 3 ->E 1,2 (5) P 2, 4 RAA(3)
The Two Questions
(i) Is (5) the conclusion of the sequent?
(ii) Is (5) derived from the basic assumptions given in the sequent?

89. n ->I(m) m,n RAA(k) must be an assumption Any kind of wff
(will be the consequent)
Any kind of wff
(will be the antecedent)
m,n RAA(k)
Premises: denials of one another
Conclusion: will be the denial of assumption: k

90. Strategy n ->I(m) m,n RAA(k) must be an assumption Any kind of wff
(will be the consequent)
Any kind of wff
(will be the antecedent)
m,n RAA(k)
Premises: denials of one another
Conclusion: will be the denial of assumption: k

91. TAs may collect this assignment
Homework
Get Proofs handout (online)
Identify and Solve first two ->I problems on handout.
Solve S14* : ~P -> Q, ~Q ⊢ P
Read pp.28-9 "double turnstile“
Study this presentation at home
esp. S14
All 10 rules committed to memory!!!
TAs may collect this assignment