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Logic as a Tool Chapter 2: Deductive Reasoning in Propositional Logic 2.4 Propositional Natural Deduction Valentin Goranko Stockholm University November 2020 Goranko Natural Deduction Natural Deduction (ND): System for structured logical

  1. Logic as a Tool Chapter 2: Deductive Reasoning in Propositional Logic 2.4 Propositional Natural Deduction Valentin Goranko Stockholm University November 2020 Goranko

  2. Natural Deduction ◮ Natural Deduction (ND): System for structured logical derivation from a set of assumptions, based on rules, specific to the logical connectives. Goranko

  3. Natural Deduction ◮ Natural Deduction (ND): System for structured logical derivation from a set of assumptions, based on rules, specific to the logical connectives. ◮ For each logical connective: introduction rules and elimination rules. Goranko

  4. Natural Deduction ◮ Natural Deduction (ND): System for structured logical derivation from a set of assumptions, based on rules, specific to the logical connectives. ◮ For each logical connective: introduction rules and elimination rules. ◮ Introduction (opening) and cancelation ( closing, discharge) of assumptions. Goranko

  5. Natural Deduction ◮ Natural Deduction (ND): System for structured logical derivation from a set of assumptions, based on rules, specific to the logical connectives. ◮ For each logical connective: introduction rules and elimination rules. ◮ Introduction (opening) and cancelation ( closing, discharge) of assumptions. ◮ Assumptions can be re-used many times before canceled. Goranko

  6. Natural Deduction ◮ Natural Deduction (ND): System for structured logical derivation from a set of assumptions, based on rules, specific to the logical connectives. ◮ For each logical connective: introduction rules and elimination rules. ◮ Introduction (opening) and cancelation ( closing, discharge) of assumptions. ◮ Assumptions can be re-used many times before canceled. ◮ Cancelation of assumptions: only when the rules allow it, but not an obligation. Goranko

  7. Natural Deduction ◮ Natural Deduction (ND): System for structured logical derivation from a set of assumptions, based on rules, specific to the logical connectives. ◮ For each logical connective: introduction rules and elimination rules. ◮ Introduction (opening) and cancelation ( closing, discharge) of assumptions. ◮ Assumptions can be re-used many times before canceled. ◮ Cancelation of assumptions: only when the rules allow it, but not an obligation. ◮ All open assumptions at the end of the derivation must be declared. Goranko

  8. Natural Deduction ◮ Natural Deduction (ND): System for structured logical derivation from a set of assumptions, based on rules, specific to the logical connectives. ◮ For each logical connective: introduction rules and elimination rules. ◮ Introduction (opening) and cancelation ( closing, discharge) of assumptions. ◮ Assumptions can be re-used many times before canceled. ◮ Cancelation of assumptions: only when the rules allow it, but not an obligation. ◮ All open assumptions at the end of the derivation must be declared. NB: the fewer (or, weaker) are the assumptions, the stronger is the claim of the derivation. Goranko

  9. ND rules for the propositional connectives Introduction rules: Elimination rules: Goranko

  10. ND rules for the propositional connectives Introduction rules: Elimination rules: A , B ( ∧ I ) A ∧ B Goranko

  11. ND rules for the propositional connectives Introduction rules: Elimination rules: A ∧ B A ∧ B A , B ( ∧ E ) ( ∧ I ) A B A ∧ B Goranko

  12. ND rules for the propositional connectives Introduction rules: Elimination rules: A ∧ B A ∧ B A , B ( ∧ E ) ( ∧ I ) A B A ∧ B A B ( ∨ I ) A ∨ B A ∨ B Goranko

  13. ND rules for the propositional connectives Introduction rules: Elimination rules: A ∧ B A ∧ B A , B ( ∧ E ) ( ∧ I ) A B A ∧ B A B ( ∨ I ) [ A ] [ B ] A ∨ B A ∨ B . . . . . . A ∨ B C C ( ∨ E ) C Goranko

  14. Introduction rules: Elimination rules: Goranko

  15. Introduction rules: Elimination rules: [ A ] . . . B ( → I ) A → B Goranko

  16. Introduction rules: Elimination rules: [ A ] A , A → B ( → E ) . . B . B ( → I ) A → B Goranko

  17. Introduction rules: Elimination rules: [ A ] A , A → B ( → E ) . . B . B ( → I ) A → B [ A ] . . . ⊥ ( ¬ I ) ¬ A Goranko

  18. Introduction rules: Elimination rules: [ A ] A , A → B ( → E ) . . B . B ( → I ) A → B A , ¬ A ( ¬ E ) ⊥ [ A ] . . . ⊥ ( ¬ I ) ¬ A Goranko

  19. Two more ND rules Ex falso quodlibet: Reductio ad absurdum: Goranko

  20. Two more ND rules Ex falso quodlibet: Reductio ad absurdum: ⊥ ( ⊥ ) A Goranko

  21. Two more ND rules Ex falso quodlibet: Reductio ad absurdum: [ ¬ A ] ⊥ ( ⊥ ) . . . A ⊥ ( RA ) A Goranko

  22. Propositional Natural Deduction: Example 1 Goranko

  23. Propositional Natural Deduction: Example 1 A ∧ B ⊢ ND B ∧ A : Goranko

  24. Propositional Natural Deduction: Example 1 A ∧ B ⊢ ND B ∧ A : A ∧ B Goranko

  25. Propositional Natural Deduction: Example 1 A ∧ B ⊢ ND B ∧ A : ( ∧ E ) A ∧ B B Goranko

  26. Propositional Natural Deduction: Example 1 A ∧ B ⊢ ND B ∧ A : ( ∧ E ) A ∧ B A ∧ B B Goranko

  27. Propositional Natural Deduction: Example 1 A ∧ B ⊢ ND B ∧ A : ( ∧ E ) A ∧ B ( ∧ E ) A ∧ B B A Goranko

  28. Propositional Natural Deduction: Example 1 A ∧ B ⊢ ND B ∧ A : ( ∧ E ) A ∧ B ( ∧ E ) A ∧ B B A ( ∧ I ) B ∧ A Goranko

  29. Propositional Natural Deduction: Examples 2 ⊢ ND A → ¬¬ A : Goranko

  30. Propositional Natural Deduction: Examples 2 ⊢ ND A → ¬¬ A : A Goranko

  31. Propositional Natural Deduction: Examples 2 ⊢ ND A → ¬¬ A : A ¬ A Goranko

  32. Propositional Natural Deduction: Examples 2 ⊢ ND A → ¬¬ A : A ¬ A ( ¬ E ) ⊥ Goranko

  33. Propositional Natural Deduction: Examples 2 ⊢ ND A → ¬¬ A : A ¬ A ( ¬ E ) ⊥ ( ¬ I ) ¬¬ A Goranko

  34. Propositional Natural Deduction: Examples 2 ⊢ ND A → ¬¬ A : [ ¬ A ] 1 A ( ¬ E ) ⊥ ( ¬ I ) 1 ¬¬ A Goranko

  35. Propositional Natural Deduction: Examples 2 ⊢ ND A → ¬¬ A : [ ¬ A ] 1 A ( ¬ E ) ⊥ ( ¬ I ) 1 ¬¬ A ( → I ) A → ¬¬ A Goranko

  36. Propositional Natural Deduction: Examples 2 ⊢ ND A → ¬¬ A : ( ¬ E ) [ A ] 2 , [ ¬ A ] 1 ⊥ ( ¬ I ) 1 ¬¬ A ( → I ) 2 A → ¬¬ A Goranko

  37. Exercise: ⊢ ND ¬¬ A → A Goranko

  38. Propositional Natural Deduction: Examples 3 A → B ⊢ ND ¬ B → ¬ A : Goranko

  39. Propositional Natural Deduction: Examples 3 A → B ⊢ ND ¬ B → ¬ A : A → B Goranko

  40. Propositional Natural Deduction: Examples 3 A → B ⊢ ND ¬ B → ¬ A : A , A → B Goranko

  41. Propositional Natural Deduction: Examples 3 A → B ⊢ ND ¬ B → ¬ A : A , A → B B Goranko

  42. Propositional Natural Deduction: Examples 3 A → B ⊢ ND ¬ B → ¬ A : A , A → B , ¬ B B Goranko

  43. Propositional Natural Deduction: Examples 3 A → B ⊢ ND ¬ B → ¬ A : A , A → B , ¬ B B ⊥ Goranko

  44. Propositional Natural Deduction: Examples 3 A → B ⊢ ND ¬ B → ¬ A : [ A ] 1 , A → B , ¬ B B ⊥ 1 ¬ A Goranko

  45. Propositional Natural Deduction: Examples 3 A → B ⊢ ND ¬ B → ¬ A : [ A ] 1 , A → B , [ ¬ B ] 2 B ⊥ 1 ¬ A 2 ¬ B → ¬ A Goranko

  46. Propositional Natural Deduction: Examples 3 A → B ⊢ ND ¬ B → ¬ A : [ A ] 1 , A → B , [ ¬ B ] 2 B ⊥ 1 ¬ A 2 ¬ B → ¬ A ¬ B → ¬ A ⊢ ND A → B : Goranko

  47. Propositional Natural Deduction: Examples 3 A → B ⊢ ND ¬ B → ¬ A : [ A ] 1 , A → B , [ ¬ B ] 2 B ⊥ 1 ¬ A 2 ¬ B → ¬ A ¬ B → ¬ A ⊢ ND A → B : [ ¬ B ] 1 , ¬ B → ¬ A , [ A ] 2 [ ¬ A ] ⊥ 1 B ( → I ) 2 A → B Goranko

  48. Propositional Natural Deduction: Examples 4 A ∨ B ⊢ ND ¬ A → B : Goranko

  49. Propositional Natural Deduction: Examples 4 A ∨ B ⊢ ND ¬ A → B : A ∨ B Goranko

  50. Propositional Natural Deduction: Examples 4 A ∨ B ⊢ ND ¬ A → B : A A ∨ B Goranko

  51. Propositional Natural Deduction: Examples 4 A ∨ B ⊢ ND ¬ A → B : ¬ A , A A ∨ B Goranko

  52. Propositional Natural Deduction: Examples 4 A ∨ B ⊢ ND ¬ A → B : ¬ A , A ⊥ A ∨ B Goranko

  53. Propositional Natural Deduction: Examples 4 A ∨ B ⊢ ND ¬ A → B : ¬ A , A ⊥ B A ∨ B Goranko

  54. Propositional Natural Deduction: Examples 4 A ∨ B ⊢ ND ¬ A → B : [ ¬ A ] 1 , A ⊥ B 1 A ∨ B ¬ A → B Goranko

  55. Propositional Natural Deduction: Examples 4 A ∨ B ⊢ ND ¬ A → B : [ ¬ A ] 1 , A ⊥ B B 1 A ∨ B ¬ A → B Goranko

  56. Propositional Natural Deduction: Examples 4 A ∨ B ⊢ ND ¬ A → B : [ ¬ A ] 1 , A ⊥ B ¬ A B 1 A ∨ B ¬ A → B Goranko

  57. Propositional Natural Deduction: Examples 4 A ∨ B ⊢ ND ¬ A → B : [ ¬ A ] 1 , A ⊥ [ ¬ A ] 2 , B B 1 2 A ∨ B ¬ A → B ¬ A → B Goranko

  58. Propositional Natural Deduction: Examples 4 A ∨ B ⊢ ND ¬ A → B : [ ¬ A ] 1 , [ A ] 3 ⊥ [ ¬ A ] 2 , [ B ] 3 B 1 2 A ∨ B ¬ A → B ¬ A → B 3 ¬ A → B Goranko

  59. Propositional Natural Deduction: Examples 5 Goranko

  60. Propositional Natural Deduction: Examples 5 ⊢ ND ( A → ( B → C )) → (( A ∧ B ) → C ) : Goranko

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