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Introductory Logic PHI 120

Presentation: “ Theorems ". Introductory Logic PHI 120. Changed Presentation. Homework. Homework over Break ( a) S1 - S27, T1 - T4 (from book ) R . Smith Guides (available online) " Proofs without tears " " Proofs with even fewer tears “ Study class presentations.

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Introductory Logic PHI 120

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  1. Presentation: “Theorems" Introductory LogicPHI 120 Changed Presentation

  2. Homework • Homework over Break • (a) S1 - S27, T1 - T4 (from book) • R. Smith Guides (available online) • "Proofs without tears" • "Proofs with even fewer tears“ • Study class presentations

  3. 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

  4. 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

  5. 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)

  6. 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)

  7. 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

  8. 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

  9. 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

  10. 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)

  11. 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

  12. 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?

  13. 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!!!!

  14. 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!!!! Either ->I or RAA Conclusion not an ->

  15. 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 Strategy of RAA 1. Assume the denial of the conclusion 2. Derive a contradiction 3. Use RAA to deny/discharge an assumption

  16. 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

  17. 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

  18. 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

  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 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

  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 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

  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 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

  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 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

  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 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

  24. P v Q ⊣⊢ ~P -> Q Multiple RAA Problems Discharge the RAA assumption last 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 yet [5] • Which assumption should you discharge first? • 1, 2, or 5

  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 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(2)

  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 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) P 4,5 RAA(2)

  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 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 1,5 (6) P 4,5 RAA(2)

  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(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 1,5 (6) P 4,5 RAA(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(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 1,5 (6) P 4,5 RAA(2) (7) ??

  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(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 1,5 (6) P 4,5 RAA(2) 1,5 (7) P v Q6 vI Look at your assumptions Is the final line the main conclusion? Are the assumptions correct on this final line?

  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(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 1,5 (6) P 4,5 RAA(2) 1,5 (7) P v Q6 vI

  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(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 1,5 (6) P 4,5 RAA(2) 1,5 (7) P v Q 6 vI (8) ??

  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(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 1,5 (6) P 4,5 RAA(2) 1,5 (7) P v Q 6 vI (8) 5,7 RAA(5) now [5]

  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(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 1,5 (6) P 4,5 RAA(2) 1,5 (7) P v Q 6 vI (8) P v Q5,7 RAA(5)

  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(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 1,5 (6) P 4,5 RAA(2) 1,5 (7) P v Q 6 vI (8) P v Q 5,7 RAA(5)

  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 (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 1,5 (6) P 4,5 RAA(2) 1,5 (7) P v Q 6 vI 1 (8) P v Q 5,7 RAA(5)

  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 1,2 (4) P v Q 3 vI 5 (5) ~(P v Q) A 1,5 (6) P 4,5 RAA(2) 1,5 (7) P v Q 6 vI 1 (8) P v Q 5,7 RAA(5) Is the final line the main conclusion? Are the assumptions correct on this final line?

  38. Theorems Sentences that can be proven from an empty set of premises

  39. Sequents • A sequent contains three elements P ⊢ Q -> P

  40. Sequents • A sequent contains three elements P⊢ Q -> P Premises (basic assumptions)

  41. Sequents • A sequent contains three elements P ⊢ Q -> P Turnstile (conclusion indicator)

  42. Sequents • A sequent contains three elements P ⊢ Q -> P Conclusion

  43. Theorems • A theorem contains only two elements ⊢ P -> (Q -> P) Turnstile (conclusion indicator)

  44. Theorems • A theorem contains only two elements ⊢ P -> (Q -> P) Conclusion Remember: every proof begins with at least one assumption.

  45. Set of Theorems T1: ⊢ P->P Identity T2: ⊢ P v ~P Excluded Middle T3: ⊢ ~(P&~P) Non-Contradiction T4:* ⊢ P->(Q->P) Weakening T5:* ⊢ (P->Q) v (Q->P) Paradox of Material Implication T6: ⊢ P<->~~P Double Negation T7: ⊢ (P<->Q)<->(Q<->P)

  46. Weakening ⊢ P -> (Q -> P) Every proof begins with at least one assumption.

  47. Weakening ⊢ P -> (Q -> P) 1 (1) ?? A Every proof begins with at least one assumption.

  48. Weakening ⊢ P -> (Q -> P) 1 (1) ?? A Strategy of -> I Assume the antecedent of the conclusion Solve for the consequent Apply ->I rule

  49. Weakening ⊢ P -> (Q -> P) 1 (1) P A Strategy of -> I Assume the antecedent of the conclusion Solve for the consequent Apply ->I rule

  50. Weakening (2) ⊢ P -> (Q -> P) 1 (1) P A Strategy of -> I Assume the antecedent of the conclusion Solve for the consequent Apply ->I rule

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