1. Introductory Logic PHI 120
Strategies for Solving Complex Problems
Introductory Logic PHI 120

2. Homework Proofs Handout: Problem Set A: S1 - S27
Problem Set B: T1 – T4
Problem Set C: S34 - S49
Problem Set D: T5 - T7

3. Homework Proofs Handout: Problem Set A: S1 - S27
Have already assigned all of these!
Proofs Handout:
Problem Set A: S1 - S27
Problem Set B: T1 – T4
Problem Set C: S34 - S49
Problem Set D: T5 - T7

4. Homework
Solve
S36 – S38 (Comm)
T5* (PMI)

5. Both sites recommended (again)
External Sites
Both sites recommended (again)
“Proofs without tears”
“Proofs with even fewer tears”

6. P <-> Q ⊣⊢ Q <-> P
Mathematical Concepts
Commutativity
P & Q ⊣⊢ Q & P
P v Q ⊣⊢ Q v P
P <-> Q ⊣⊢ Q <-> P
A simple problem – to highlight strategy

7. P<->Q ⊣⊢ Q<->P

8. P<->Q ⊣⊢ Q<->P
Q <-> P ⊢ P <-> Q

9. P<->Q ⊣⊢ Q<->P
Think <->I (why?)
P <-> Q ⊢ Q <-> P
(1) P <-> Q A
(2)
Q <-> P ⊢ P <-> Q
Think <->E (why?)
Φ <-> Ψ
is equivalent to
(Φ -> Ψ) & (Ψ -> Φ)

10. P<->Q ⊣⊢ Q<->P
(1) P <-> Q A
1 (2) P -> Q <->E
Q <-> P ⊢ P <-> Q
Φ <-> Ψ
is equivalent to
(Φ -> Ψ) & (Ψ -> Φ)

11. P<->Q ⊣⊢ Q<->P
(1) P <-> Q A
1 (2) P -> Q <->E
1 (3) Q -> P <->E
Q <-> P ⊢ P <-> Q
Φ <-> Ψ
is equivalent to
(Φ -> Ψ) & (Ψ -> Φ)

12. P<->Q ⊣⊢ Q<->P
(1) P <-> Q A
1 (2) P -> Q <->E
1 (3) Q -> P <->E
(4) ??
Q <-> P ⊢ P <-> Q
Φ <-> Ψ
is equivalent to
(Φ -> Ψ) & (Ψ -> Φ)

13. P<->Q ⊣⊢ Q<->P
(1) P <-> Q A
1 (2) P -> Q <->E
1 (3) Q -> P <->E
(4) ,3 <->I
Q <-> P ⊢ P <-> Q
Φ <-> Ψ
is equivalent to
(Φ -> Ψ) & (Ψ -> Φ)

14. P<->Q ⊣⊢ Q<->P
(1) P <-> Q A
1 (2) P -> Q <->E
1 (3) Q -> P <->E
(4) Q <-> P ,3 <->I
Q <-> P ⊢ P <-> Q
Φ <-> Ψ
is equivalent to
(Φ -> Ψ) & (Ψ -> Φ)

15. P<->Q ⊣⊢ Q<->P
(1) P <-> Q A
1 (2) P -> Q <->E
1 (3) Q -> P <->E
(4) Q <-> P ,3 <->I
Q <-> P ⊢ P <-> Q
Φ <-> Ψ
is equivalent to
(Φ -> Ψ) & (Ψ -> Φ)

16. P<->Q ⊣⊢ Q<->P
(1) P <-> Q A
1 (2) P -> Q <->E
1 (3) Q -> P <->E
1 (4) Q <-> P ,3 <->I
Q <-> P ⊢ P <-> Q
Φ <-> Ψ
is equivalent to
(Φ -> Ψ) & (Ψ -> Φ)

17. P<->Q ⊣⊢ Q<->P
P <-> Q ⊢ Q <-> P 1 (1) P <-> Q A 1 (2) P -> Q 1 <->E 1 (3) Q -> P 1 <->E 1 (4) Q <-> P 2,3 <->I
Q <-> P ⊢ P <-> Q
Φ <-> Ψ
is equivalent to
(Φ -> Ψ) & (Ψ -> Φ)
Easily solvable IF:
You understand <->I
You understand <->E

18. T5: ⊢ (P -> Q) v (Q -> P)
Complex Theorem
2 x 2 Question Method
T5: ⊢ (P -> Q) v (Q -> P)

19. ⊢ (P -> Q) v (Q -> P)
Formulate a strategy!

20. ⊢ (P -> Q) v (Q -> P)
1 (1) P A (2) ?
Formulate a strategy!
Solve for consequent, Q

21. ⊢ (P -> Q) v (Q -> P)
1 (1) P A 2 (2) Q A (3) ?

22. ⊢ (P -> Q) v (Q -> P)
1 (1) P A 2 (2) Q A 2 (3) P -> Q 2 ->I(1)

23. ⊢ (P -> Q) v (Q -> P)
1 (1) P A 2 (2) Q A 2 (3) P -> Q 2 ->I(1) 2 (4) (P -> Q) v (Q -> P) 3vI
Recognize the problem!
Now build the wedge

24. ⊢ (P -> Q) v (Q -> P)
1 (1) P A 2 (2) Q A 2 (3) P -> Q 2 ->I(1) 2 (4) (P -> Q) v (Q -> P) 3vI 5 (5) ~((P -> Q) v (Q -> P)) A
Begin RAA Strategy

25. ⊢ (P -> Q) v (Q -> P)
1 (1) P A 2 (2) Q A 2 (3) P -> Q 2 ->I(1) 2 (4) (P -> Q) v (Q -> P) 3vI 5 (5) ~((P -> Q) v (Q -> P)) A 5 (6) ~Q 4,5 RAA(2)
Why discharge (2)?

26. ⊢ (P -> Q) v (Q -> P)
1 (1) P A 2 (2) Q A 2 (3) P -> Q 2 ->I(1) 2 (4) (P -> Q) v (Q -> P) 3vI 5 (5) ~((P -> Q) v (Q -> P)) A 5 (6) ~Q 4,5 RAA(2)

27. ⊢ (P -> Q) v (Q -> P)
1 (1) P A 2 (2) Q A 2 (3) P -> Q 2 ->I(1) 2 (4) (P -> Q) v (Q -> P) 3vI 5 (5) ~((P -> Q) v (Q -> P)) A 5 (6) ~Q 4,5 RAA(2)

28. ⊢ (P -> Q) v (Q -> P)
1 (1) P A 2 (2) Q A 2 (3) P -> Q 2 ->I(1) 2 (4) (P -> Q) v (Q -> P) 3vI 5 (5) ~((P -> Q) v (Q -> P)) A 5 (6) ~Q 4,5 RAA(2)
(nested strategy)
Make second disjunct (Q -> P)

29. ⊢ (P -> Q) v (Q -> P)
1 (1) P A 2 (2) Q A 2 (3) P -> Q 2 ->I(1) 2 (4) (P -> Q) v (Q -> P) 3vI 5 (5) ~((P -> Q) v (Q -> P)) A 5 (6) ~Q 4,5 RAA(2)
(nested strategy)
Make second disjunct (Q -> P)

30. ⊢ (P -> Q) v (Q -> P)
1 (1) P A 2 (2) Q A 2 (3) P -> Q 2 ->I(1) 2 (4) (P -> Q) v (Q -> P) 3vI 5 (5) ~((P -> Q) v (Q -> P)) A 5 (6) ~Q 4,5 RAA(2) 7 (7) Q A
(nested strategy)
Make second disjunct (Q -> P)

31. ⊢ (P -> Q) v (Q -> P)
1 (1) P A 2 (2) Q A 2 (3) P -> Q 2 ->I(1) 2 (4) (P -> Q) v (Q -> P) 3vI 5 (5) ~((P -> Q) v (Q -> P)) A 5 (6) ~Q 4,5 RAA(2) 7 (7) Q A 8 (8) ~P A
(nested strategy)
Make second disjunct (Q -> P)
Use RAA to deny/discharge an assumption

32. ⊢ (P -> Q) v (Q -> P)
1 (1) P A 2 (2) Q A 2 (3) P -> Q 2 ->I(1) 2 (4) (P -> Q) v (Q -> P) 3vI 5 (5) ~((P -> Q) v (Q -> P)) A 5 (6) ~Q 4,5 RAA(2) 7 (7) Q A 8 (8) ~P A 5,7 (9) P 6,7 RAA(8)
(nested strategy)
Make second disjunct (Q -> P)
Use RAA to deny/discharge an assumption

33. ⊢ (P -> Q) v (Q -> P)
1 (1) P A 2 (2) Q A 2 (3) P -> Q 2 ->I(1) 2 (4) (P -> Q) v (Q -> P) 3vI 5 (5) ~((P -> Q) v (Q -> P)) A 5 (6) ~Q 4,5 RAA(2) 7 (7) Q A 8 (8) ~P A 5,7 (9) P 6,7 RAA(8) 5 (10) Q -> P 9 ->I(7)
(nested strategy)
Make second disjunct (Q -> P)

34. ⊢ (P -> Q) v (Q -> P)
1 (1) P A 2 (2) Q A 2 (3) P -> Q 2 ->I(1) 2 (4) (P -> Q) v (Q -> P) 3vI 5 (5) ~((P -> Q) v (Q -> P)) A 5 (6) ~Q 4,5 RAA(2) 7 (7) Q A 8 (8) ~P A 5,7 (9) P 6,7 RAA(8) 5 (10) Q -> P 9 ->I(7) 5 (11) (P -> Q) v (Q -> P) 10 vI
(nested strategy)
Recognize the problem!

35. ⊢ (P -> Q) v (Q -> P)
1 (1) P A 2 (2) Q A 2 (3) P -> Q 2 ->I(1) 2 (4) (P -> Q) v (Q -> P) 3vI 5 (5) ~((P -> Q) v (Q -> P)) A 5 (6) ~Q 4,5 RAA(2) 7 (7) Q A 8 (8) ~P A 5,7 (9) P 6,7 RAA(8) 5 (10) Q -> P 9 ->I(7) 5 (11) (P -> Q) v (Q -> P) 10 vI (12) (P -> Q) v (Q -> P) 5,11 RAA(5)

36. ⊢ (P -> Q) v (Q -> P)
Why is this complex?
The conclusion is a wedge: ⊢ (P -> Q) v (Q -> P)
⊢ (P -> Q) v (Q -> P)

37. ⊢ (P -> Q) v (Q -> P)
1 (1) P A 2 (2) Q A 2 (3) P -> Q 2 ->I(1) 2 (4) (P -> Q) v (Q -> P) 3vI 5 (5) ~((P -> Q) v (Q -> P)) A 5 (6) ~Q 4,5 RAA(2) 7 (7) Q A 8 (8) ~P A 5,7 (9) P 6,7 RAA(8) 5 (10) Q -> P 9 ->I(7) 5 (11) (P -> Q) v (Q -> P) 10 vI (12) (P -> Q) v (Q -> P) 5,11 RAA(5)

38. ⊢ (P -> Q) v (Q -> P)
1 (1) P A 2 (2) Q A 2 (3) P -> Q 2 ->I(1) 2 (4) (P -> Q) v (Q -> P) 3vI 5 (5) ~((P -> Q) v (Q -> P)) A 5 (6) ~Q 4,5 RAA(2) 7 (7) Q A 8 (8) ~P A 5,7 (9) P 6,7 RAA(8) 5 (10) Q -> P 9 ->I(7) 5 (11) (P -> Q) v (Q -> P) 10 vI (12) (P -> Q) v (Q -> P) 5,11 RAA(5)

39. ⊢ (P -> Q) v (Q -> P)
Why was this proof difficult?
Built the arrow
Built the wedge
Added problem – too many assumptions!
Complex Proofs require nested strategies
overlapping strategies
proceed step by step
->I vI
RAA

40. Homework
Solve
S36 – S38 (Comm)
T5* (PMI)