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Introductory Logic PHI 120

Complex Problems II. Introductory Logic PHI 120. REMINDER. Grading, attendance, accommodation issues Dealt with by your RECITATION INSTRUCTOR. Homework. Proofs Handout: Problem Set A: S1 - S27 Problem Set B: T1 – T4 Problem Set C: S34 - S49 Problem Set D: T5 - T7. Homework.

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Introductory Logic PHI 120

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  1. Complex Problems II Introductory LogicPHI 120

  2. REMINDER Grading, attendance, accommodation issues Dealt with by your RECITATION INSTRUCTOR

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

  4. Homework • (1) Solve • S39 – S49from Proofs Handout • T5 – T7 • Come to review with specific problems • (3) Study Guide 2

  5. Mathematical Concepts-- Commutativity -- -- Distribution ---- Associativity -- Problems from Ex. 1.5.1

  6. Mathematical Concepts Commutativity P & Q⊣⊢ Q & P 2 x 2 Question Method P v Q ⊣⊢ Q v P P <-> Q ⊣⊢ Q <-> P

  7. PvQ ⊣⊢ QvP

  8. PvQ ⊣⊢ QvP P v Q ⊢ Q v P Q v P ⊢ P v Q

  9. PvQ ⊣⊢ QvP Think vI (why?) P v Q ⊢ Q v P • (1) P v Q A (2) Q v P ⊢ P v Q Think vE (why?) The Two Questions

  10. PvQ ⊣⊢ QvP P v Q ⊢ Q v P • (1) P v Q A (2) A Q v P ⊢ P v Q

  11. PvQ ⊣⊢ QvP P v Q ⊢ Q v P • (1) P v Q A (2) ~P A Q v P ⊢ P v Q

  12. PvQ ⊣⊢ QvP P v Q ⊢ Q v P • (1) P v Q A 2 (2) ~P A Q v P ⊢ P v Q

  13. PvQ ⊣⊢ QvP P v Q ⊢ Q v P • (1) P v Q A • (2) ~P A (3) Q v P ⊢ P v Q

  14. PvQ ⊣⊢ QvP P v Q ⊢ Q v P • (1) P v Q A • (2) ~P A (3) 1,2 vE Q v P ⊢ P v Q

  15. PvQ ⊣⊢ QvP P v Q ⊢ Q v P • (1) P v Q A • (2) ~P A (3) Q 1,2 vE Q v P ⊢ P v Q

  16. PvQ ⊣⊢ QvP P v Q ⊢ Q v P • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE Q v P ⊢ P v Q

  17. PvQ ⊣⊢ QvP P v Q ⊢ Q v P • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE (4) Q v P ⊢ P v Q

  18. PvQ ⊣⊢ QvP vI P v Q ⊢ Q v P • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE (4) ?? Q v P ⊢ P v Q

  19. PvQ ⊣⊢ QvP vI P v Q ⊢ Q v P • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE (4) 3 vI Q v P ⊢ P v Q

  20. PvQ ⊣⊢ QvP vI P v Q ⊢ Q v P • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE (4) Q v P 3 vI Q v P ⊢ P v Q

  21. PvQ ⊣⊢ QvP vI P v Q ⊢ Q v P • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI Q v P ⊢ P v Q The Two Questions Either ->I or RAA

  22. PvQ ⊣⊢ QvP P v Q ⊢ Q v P • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI (5) Q v P ⊢ P v Q RAA Strategy

  23. PvQ ⊣⊢ QvP P v Q ⊢ Q v P • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI (5) A Q v P ⊢ P v Q RAA Strategy

  24. PvQ ⊣⊢ QvP P v Q ⊢ Q v P • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI (5) ~(Q v P) A Q v P ⊢ P v Q RAA Strategy

  25. PvQ ⊣⊢ QvP P v Q ⊢ Q v P • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI 5(5) ~(Q v P) A Q v P ⊢ P v Q RAA Strategy Remember: assumptions take the number of the line

  26. PvQ ⊣⊢ QvP P v Q ⊢ Q v P • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI 5(5) ~(Q v P) A Q v P ⊢ P v Q

  27. PvQ ⊣⊢ QvP P v Q ⊢ Q v P • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI • (5) ~(Q v P) A (6) Q v P ⊢ P v Q

  28. PvQ ⊣⊢ QvP P v Q ⊢ Q v P • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI • (5) ~(Q v P) A (6) 4,5 RAA(?) Q v P ⊢ P v Q

  29. PvQ ⊣⊢ QvP P v Q ⊢ Q v P • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI • (5) ~(Q v P) A (6) 4,5 RAA(?) Q v P ⊢ P v Q “save the best for last”

  30. PvQ ⊣⊢ QvP P v Q ⊢ Q v P • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI • (5) ~(Q v P) A (6) 4,5 RAA(2) Q v P ⊢ P v Q

  31. PvQ ⊣⊢ QvP P v Q ⊢ Q v P • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI • (5) ~(Q v P) A (6) P 4,5 RAA(2) Q v P ⊢ P v Q

  32. PvQ ⊣⊢ QvP P v Q ⊢ Q v P • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI 5 (5) ~(Q v P) A 1,5 (6) P 4,5 RAA(2) Q v P ⊢ P v Q

  33. PvQ ⊣⊢ QvP P v Q ⊢ Q v P • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI • (5) ~(Q v P) A 1,5 (6) P 4,5 RAA(2) (7) Q v P ⊢ P v Q

  34. PvQ ⊣⊢ QvP vI P v Q ⊢ Q v P • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI • (5) ~(Q v P) A 1,5 (6) P 4,5 RAA(2) (7) ?? Q v P ⊢ P v Q

  35. PvQ ⊣⊢ QvP vI P v Q ⊢ Q v P • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI • (5) ~(Q v P) A 1,5 (6) P 4,5 RAA(2) (7) 6 vI Q v P ⊢ P v Q

  36. PvQ ⊣⊢ QvP vI P v Q ⊢ Q v P • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI • (5) ~(Q v P) A 1,5 (6) P 4,5 RAA(2) (7) Q v P 6 vI Q v P ⊢ P v Q

  37. PvQ ⊣⊢ QvP vI P v Q ⊢ Q v P • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI • (5) ~(Q v P) A 1,5 (6) P 4,5 RAA(2) 1,5 (7) Q v P 6 vI Q v P ⊢ P v Q The Two Questions

  38. PvQ ⊣⊢ QvP P v Q ⊢ Q v P • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI • (5) ~(Q v P) A 1,5 (6) P 4,5 RAA(2) 1,5 (7) Q v P 6 vI (8) Q v P ⊢ P v Q

  39. PvQ ⊣⊢ QvP P v Q ⊢ Q v P • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI • (5) ~(Q v P) A 1,5 (6) P 4,5 RAA(2) 1,5 (7) Q v P 6 vI (8) 5,7 RAA(5) Q v P ⊢ P v Q

  40. PvQ ⊣⊢ QvP P v Q ⊢ Q v P • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI • (5) ~(Q v P) A 1,5 (6) P 4,5 RAA(2) 1,5 (7) Q v P 6 vI (8) Q v P 5,7 RAA(5) Q v P ⊢ P v Q

  41. PvQ ⊣⊢ QvP P v Q ⊢ Q v P • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI 5 (5) ~(Q v P) A 1,5 (6) P 4,5 RAA(2) 1,5 (7) Q v P 6 vI (8) Q v P 5,7 RAA(5) Q v P ⊢ P v Q

  42. PvQ ⊣⊢ QvP P v Q ⊢ Q v P • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI • (5) ~(Q v P) A 1,5 (6) P 4,5 RAA(2) 1,5 (7) Q v P 6 vI 1(8) Q v P 5,7 RAA(5) Q v P ⊢ P v Q

  43. PvQ ⊣⊢ QvP P v Q ⊢ Q v P • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI • (5) ~(Q v P) A 1,5 (6) P 4,5 RAA(2) 1,5 (7) Q v P 6 vI 1 (8) Q v P 5,7 RAA(5) Q v P ⊢ P v Q Solve onyour own

  44. Mathematical Concepts Distribution P & (Q v R)⊣⊢ (P & Q) v (P & R) P v (Q & R) ⊣⊢ (P v Q) & (P v R)

  45. Mathematical Concepts Distribution P & (Q v R)⊣⊢ (P & Q) v (P & R) P v (Q & R) ⊣⊢ (P v Q) & (P v R)

  46. P&(QvR) ⊣⊢ (P&Q)v(P&R)

  47. P&(QvR) ⊣⊢ (P&Q)v(P&R) P & (Q v R) ⊢ (P & Q) v (P & R) Do this left side one on your own (P & Q) v (P & R) ⊢ P & (Q v R)

  48. P&(QvR) ⊣⊢ (P&Q)v(P&R) (P & Q) v (P & R) ⊢ P & (Q v R) 1 (1) (P & Q) v (P & R) A

  49. P&(QvR) ⊣⊢ (P&Q)v(P&R) (P & Q) v (P & R) ⊢ P & (Q v R) 1 (1) (P & Q) v (P & R) A (2) ?? Recognize the RAA! If forced to add assumptions, you may begin with the proper strategy.

  50. P&(QvR) ⊣⊢ (P&Q)v(P&R) (P & Q) v (P & R) ⊢ P & (Q v R) 1 (1) (P & Q) v (P & R) A 2 (2) ~(P & (Q v R)) A

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