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Introductory Logic PHI 120

Presentation: “ Double Turnstile Problems ". Introductory Logic PHI 120. Homework. Proofs: 1.5.1 (A/H, p.29-30) S21 – S24 (v ->) S25 – S27 (the dilemmas) S44 (Imp/Exp) External Web Pages: “ R. Smith Guide: Proofs without tears ” available through class web page. ->I and RAA.

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Introductory Logic PHI 120

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  1. Presentation: “Double Turnstile Problems" Introductory LogicPHI 120

  2. Homework • Proofs: 1.5.1 (A/H, p.29-30) • S21 – S24 (v ->) • S25 – S27 (the dilemmas) • S44 (Imp/Exp) • External Web Pages: • “R. Smith Guide: Proofs without tears” • available through class web page

  3. ->I and RAA

  4. Internalize These Strategies ->I • Assume antecedent of the conclusion • Solve for the consequent • Apply ->I rule RAA • Assume the denial of what you’re solving for • Derive a contradiction • Apply RAA rule

  5. Double Turnstile Problems P v Q ⊣⊢ ~P -> Q

  6. P v Q ⊣⊢ ~P -> Q

  7. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q ~P -> Q ⊢ P v Q

  8. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q ~P -> Q ⊢ P v Q

  9. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A (2) ~P -> Q ⊢ P v Q

  10. P v Q ⊣⊢ ~P -> Q P v Q⊢ ~P -> Q • (1) P v Q A (2) ?? ~P -> Q ⊢ P v Q

  11. P v Q ⊣⊢ ~P -> Q P v Q⊢ ~P ->Q • (1) P v Q A (2) ?? ~P -> Q ⊢ P v Q

  12. P v Q ⊣⊢ ~P -> Q P v Q⊢ ~P-> Q • (1) P v Q A (2) ?? ~P -> Q ⊢ P v Q

  13. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A (2) ?? ~P -> Q ⊢ P v Q Strategy of ->I 1. Assume the antecedent of the conclusion 2. Solve for the consequent (as a conclusion) 3. Apply ->I rule.

  14. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A 2 (2) ~PA ~P -> Q ⊢ P v Q Strategy of ->I 1. Assume the antecedent of the conclusion 2. Solve for the consequent (as a conclusion) 3. Apply ->I rule.

  15. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P-> Q • (1) P v Q A 2 (2) ~P A ~P -> Q ⊢ P v Q We now have too many assumptions! Strategy of ->I 1. Assume the antecedent of the conclusion 2. Solve for the consequent (as a conclusion) 3. Apply ->I rule.

  16. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A (3) ?? ~P -> Q ⊢ P v Q Phase II: Solve for consequent Strategy of ->I 1. Assume the antecedent of the conclusion 2. Solve for the consequent (as a conclusion) 3. Apply ->I rule.

  17. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A (3) ?? ~P -> Q ⊢ P v Q Strategy of ->I 1. Assume the antecedent of the conclusion 2. Solve for the consequent (as a conclusion) 3. Apply ->I rule.

  18. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A (3) Q1,2vE ~P -> Q ⊢ P v Q Strategy of ->I 1. Assume the antecedent of the conclusion 2. Solve for the consequent (as a conclusion) 3. Apply ->I rule.

  19. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE ~P -> Q ⊢ P v Q Strategy of ->I 1. Assume the antecedent of the conclusion 2. Solve for the consequent (as a conclusion) 3. Apply ->I rule.

  20. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE (4) ?? ~P -> Q ⊢ P v Q Phase III: Apply ->I rule Strategy of ->I 1. Assume the antecedent of the conclusion 2. Solve for the consequent (as a conclusion) 3. Apply ->I rule.

  21. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE (4) ~P -> Q3 ->I(2) ~P -> Q ⊢ P v Q

  22. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q

  23. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q

  24. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q

  25. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q Is the final line the main conclusion? Are the assumptions correct on this final line?

  26. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q

  27. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q 1 (1) ~P -> Q A

  28. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q • (1) ~P -> Q A (2)

  29. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q • (1) ~P -> Q A (2)

  30. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q • (1) ~P -> Q A (2) ?? Look at the premise in relation to the conclusion?

  31. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q • (1) ~P -> Q A (2) ?? Look at the premise in relation to the conclusion?

  32. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q • (1) ~P -> Q A (2) A Assume what?

  33. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q • (1) ~P -> Q A 2 (2) ~P A The antecedent of (1)

  34. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q • (1) ~P -> Q A • (2) ~P A (3)

  35. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q • (1) ~P -> Q A • (2) ~P A (3) Q 1,2 ->E

  36. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q • (1) ~P -> Q A • (2) ~P A (3) Q 1,2 ->E

  37. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q 1 (1) ~P -> Q A 2 (2) ~P A 1,2 (3) Q 1,2 ->E

  38. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q 1(1) ~P -> Q A 2(2) ~P A 1,2 (3) Q 1,2 ->E (4) ?? Make the wedge (i.e., the conclusion)

  39. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q 1(1) ~P -> Q A 2(2) ~P A 1,2 (3) Q 1,2 ->E 1,2 (4) P v Q 3 vI

  40. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q 1(1) ~P -> Q A 2(2) ~P A 1,2 (3) Q 1,2 ->E 1,2 (4) P v Q 3 vI Is the final line the main conclusion? Are the assumptions correct on this final line?

  41. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q 1(1) ~P -> Q A 2(2) ~P A 1,2 (3) Q 1,2 ->E 1,2 (4) P v Q 3 vI Too many assumptions!!!!

  42. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q 1(1) ~P -> Q A 2(2) ~P A 1,2 (3) Q 1,2 ->E 1,2 (4) P v Q 3 vI To discharge assumptions: ->I or RAA?

  43. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q 1(1) ~P -> Q A 2(2) ~P A 1,2 (3) Q 1,2 ->E 1,2 (4) P v Q 3 vI (5) A Strategy of RAA 1. Assume the denial of the conclusion 2. Derive a contradiction 3. Use RAA to deny/discharge an assumption

  44. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q 1(1) ~P -> Q A 2(2) ~P A 1,2 (3) Q 1,2 ->E 1,2 (4) P v Q 3 vI (5) ~(P v Q)A Strategy of RAA 1. Assume the denial of the conclusion 2. Derive a contradiction 3. Use RAA to deny/discharge an assumption

  45. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q 1(1) ~P -> Q A 2(2) ~P A 1,2 (3) Q 1,2 ->E 1,2 (4) P v Q 3 vI 5 (5) ~(P v Q)A Strategy of RAA 1. Assume the denial of the conclusion 2. Derive a contradiction 3. Use RAA to deny/discharge an assumption

  46. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q 1(1) ~P -> Q A 2(2) ~P A 1,2 (3) Q 1,2 ->E 1,2 (4) P v Q 3 vI 5 (5) ~(P v Q) A Strategy of RAA 1. Assume the denial of the conclusion 2. Derive a contradiction 3. Use RAA to deny/discharge an assumption

  47. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q 1(1) ~P -> Q A 2(2) ~P A 1,2 (3) Q 1,2 ->E 1,2 (4) P v Q 3 vI • (5) ~(P v Q) A (6) Strategy of RAA 1. Assume the denial of the conclusion 2. Derive a contradiction 3. Use RAA to deny/discharge an assumption

  48. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q 1 (1) ~P -> Q A 2 (2) ~P A 1,2 (3) Q 1,2 ->E 1,2 (4) P v Q 3 vI 5 (5) ~(P v Q) A (6) 4,5 RAA(?) Strategy of RAA 1. Assume the denial of the conclusion 2. Derive a contradiction 3. Use RAA to deny/discharge an assumption

  49. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q 1 (1) ~P -> Q A 2 (2) ~P A 1,2 (3) Q 1,2 ->E 1,2 (4) P v Q 3 vI 5 (5) ~(P v Q) A (6) 4,5 RAA(?) Assumptions • Which assumption should you discharge first? • 1, 2, or 5

  50. P v Q ⊣⊢ ~P -> Q P v Q ⊢ ~P -> Q • (1) P v Q A • (2) ~P A 1,2 (3) Q 1,2 vE 1 (4) ~P -> Q 3 ->I(2) ~P -> Q ⊢ P v Q 1(1) ~P -> Q A 2 (2) ~P A 1,2 (3) Q 1,2 ->E 1,2 (4) P v Q 3 vI 5 (5) ~(P v Q) A (6) 4,5 RAA(?) not [1] • Which assumption should you discharge first? • 1, 2, or 5

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