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Introductory Logic PHI 120

Presentation: “Basic E and I Strategies". Introductory Logic PHI 120. Homework I. Memorize the primitive rules Capable of writing the annotation Cite how many premises make up each rule Cite what kind of premises make up each rule Cite what sort of conclusion may be derived.

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Introductory Logic PHI 120

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  1. Presentation: “Basic E and I Strategies" Introductory LogicPHI 120

  2. Homework I • Memorize the primitive rules • Capable of writing the annotation • Cite how many premises make up each rule • Cite what kind of premises make up each rule • Cite what sort of conclusion may be derived Except ->I and RAA See The Rules Handout

  3. Homework I • Memorize the primitive rules • Capable of writing theannotation m, n &I • Cite how many premisesmake up each rule two premise rule • Cite what kind of premisesmake up each rule each can be any kind of wff (i.e., the 2 conjuncts) • Cite what sort of conclusionmay be derived a conjunction See The Rules Handout

  4. Homework I • Memorize the 10 primitive rules • Capable of writing the annotation m, n &I • Cite how many premisesmake up each rule two premise rule • Citewhat kind of premises make up each rule each can be any kind of wff (i.e., the 2 conjuncts)) • Cite what sort of conclusionmay bederived a conjunction See The Rules Handout

  5. Homework I • Memorize these primitive rules • Capable of writing the annotation m, n &I • Cite how many premises make up each rule two premise rule • Cite what kind of premises make up each rule each can be any kind of wff (i.e., the 2 conjuncts) • Cite what sort of conclusion may be derived a conjunction &I, vI, <->I &E, vE, ->E, <->E These rules should be memorized by now! See The Rules Handout

  6. Homework II To memorize the rules, you need to practice doing proofs. To practice proofs, you need to have the rules memorized • Ex. 1.4.2: S1 – S10 • First: solve problems with * • * answers in back of book • Second: all others • Seek help with your TA or me, if confused See class web page for Office Hours

  7. Review The two major kinds of rules

  8. Rules • Elimination Rules (4) • Allow you to break • Every elimination rule will have a premise of the kind indicated by that rule • Introduction Rules (4) • Allow you to make • The conclusion of every introduction rule will be the kind of statement made that rule

  9. Solving for the Conclusion Writing the Proof

  10. Writing the Proof S5: P -> Q, P -> R, P ⊢ Q & R

  11. Writing the Proof S5: P -> Q, P -> R, P ⊢ Q & R (1) A line of a proof contains four elements: (i) line number (number within parentheses) (ii) annotation (at the very right) (iii) sentence derived (next to line number) (iv) assumption set (number to very left)

  12. Writing the Proof S5: P -> Q, P -> R, P ⊢ Q & R (1) A A line of a proof contains four elements: (i) line number (number within parentheses) (ii) annotation (at the very right) (iii) sentence derived (next to line number) (iv) assumption set (number to very left)

  13. Writing the Proof S5: P -> Q, P -> R, P ⊢ Q & R (1) P -> Q A A line of a proof contains four elements: (i) line number (number within parentheses) (ii) annotation (at the very right) (iii) sentence derived (next to line number) (iv) assumption set (number to very left)

  14. Writing the Proof S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A A line of a proof contains four elements: (i) line number (number within parentheses) (ii) annotation (at the very right) (iii) sentence derived (next to line number) (iv) assumption set (number to very left)

  15. Writing the Proof S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A (2) A line of a proof contains four elements: (i) line number (number within parentheses) (ii) annotation (at the very right) (iii) sentence derived (next to line number) (iv) assumption set (number to very left)

  16. Writing the Proof S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A A line of a proof contains four elements: (i) line number (number within parentheses) (ii) annotation (at the very right) (iii) sentence derived (next to line number) (iv) assumption set (number to very left)

  17. Writing the Proof S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A (3)

  18. Writing the Proof S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A The Basic Premises

  19. S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A How to Read a Problem

  20. Writing the Proof “Q & R” is not embedded in premises! S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A Whenever solving for a conclusion, ask yourself two questions: 1) What is the conclusion? 2) How is this conclusion embedded in the premises? General Strategy EliminationorIntroduction?

  21. Writing the Proof “Q & R” is not embedded in premises! S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A General Strategy EliminationorIntroduction?

  22. Writing the Proof “Q & R” is not embedded in premises! S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A (4) Possible premise for ->E Possible premise for ->E Antecedent of (1) and (2)

  23. Writing the Proof S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A (4) m, n ->E

  24. Writing the Proof S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A (4) 1, 3 ->E

  25. Writing the Proof S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A (4) Q 1, 3 ->E

  26. Writing the Proof S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A (4) Q 1, 3 ->E

  27. Writing the Proof S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A (4) Q 1, 3 ->E

  28. Writing the Proof S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A 1,3 (4) Q 1, 3 ->E

  29. Writing the Proof S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A 1,3 (4) Q 1, 3 ->E (5)

  30. Writing the Proof S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A 1,3 (4) Q 1, 3 ->E (5) m, n ->E

  31. Writing the Proof S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A 1,3 (4) Q 1, 3 ->E (5) 2, 3 ->E

  32. Writing the Proof S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A 1,3 (4) Q 1, 3 ->E (5) R 2, 3 ->E

  33. Writing the Proof S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A 1,3 (4) Q 1, 3 ->E (5) R 2, 3 ->E

  34. Writing the Proof S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A 1,3 (4) Q 1, 3 ->E (5) R 2, 3 ->E

  35. Writing the Proof S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A 1,3 (4) Q 1, 3 ->E 2,3 (5) R 2, 3 ->E

  36. Writing the Proof S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A 1,3 (4) Q 1, 3 ->E 2,3 (5) R 2, 3 ->E (6) ??

  37. Writing the Proof S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A 1,3 (4) Q 1, 3 ->E 2,3 (5) R 2, 3 ->E (6) m, n &I

  38. Writing the Proof S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A 1,3 (4) Q 1, 3 ->E 2,3 (5) R 2, 3 ->E (6) 4, 5 &I

  39. Writing the Proof S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A 1,3 (4) Q 1, 3 ->E 2,3 (5) R 2, 3 ->E (6) Q&R 4, 5 &I

  40. Writing the Proof S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A 1,3 (4) Q 1, 3 ->E 2,3 (5) R 2, 3 ->E (6) Q & 4, 5 &I

  41. Writing the Proof S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A 1,3 (4) Q 1, 3 ->E 2,3 (5) R 2, 3 ->E (6) Q & R 4, 5 &I

  42. Writing the Proof S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A 1,3 (4) Q 1, 3 ->E 2,3 (5) R 2, 3 ->E (6) Q & R 4, 5 &I

  43. Writing the Proof S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A 1,3 (4) Q 1, 3 ->E 2,3 (5) R 2, 3 ->E (6) Q & R 4, 5 &I

  44. Writing the Proof S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A 1,3 (4) Q 1, 3 ->E 2,3 (5) R 2, 3 ->E 1,2,3 (6) Q & R 4, 5 &I

  45. Writing the Proof End every proof with two questions: Is this the main conclusion? Are the assumptions correct? S5: P -> Q, P -> R, P ⊢ Q & R 1 (1) P -> Q A 2 (2) P -> R A 3 (3) P A 1,3 (4) Q 1, 3 ->E 2,3 (5) R 2, 3 ->E 1,2,3 (6) Q & R 4, 5 &I

  46. Key Lesson 2 x 2 Questions A. Solving for a Conclusion 1) What is the conclusion? 2) How is this conclusion embedded in the premises? B. The End of a Proof • Is the final line the main conclusion? 2) Are the assumptions correct? • Strategy

  47. Homework • Memorize primitive rules(except ->I and RAA) • Ex. 1.4.2: S1 – S10 • First: solve problems with * • * answers in back of book • Second: all others To memorize the rules, you need to practice doing proofs. To practice proofs, you need to have the rules memorized

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