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Logic as a Tool Chapter 1: Understanding Propositional Logic 1.2 Propositional logical consequence Logically correct inferences Valentin Goranko Stockholm University September 2016 Goranko Propositional logical consequence A propositional

  1. Logic as a Tool Chapter 1: Understanding Propositional Logic 1.2 Propositional logical consequence Logically correct inferences Valentin Goranko Stockholm University September 2016 Goranko

  2. Propositional logical consequence A propositional formula C is a logical consequence from the propositional formulae A 1 , . . . , A n , Goranko

  3. Propositional logical consequence A propositional formula C is a logical consequence from the propositional formulae A 1 , . . . , A n , denoted A 1 , . . . , A n | = C , Goranko

  4. Propositional logical consequence A propositional formula C is a logical consequence from the propositional formulae A 1 , . . . , A n , denoted A 1 , . . . , A n | = C , if C is true whenever all A 1 , . . . , A n are true, Goranko

  5. Propositional logical consequence A propositional formula C is a logical consequence from the propositional formulae A 1 , . . . , A n , denoted A 1 , . . . , A n | = C , if C is true whenever all A 1 , . . . , A n are true, i.e., every assignment of truth-values to the variables occurring in A 1 , . . . , A n , C which renders the formulae A 1 , . . . , A n true, renders the formula C true, too. Goranko

  6. Propositional logical consequence A propositional formula C is a logical consequence from the propositional formulae A 1 , . . . , A n , denoted A 1 , . . . , A n | = C , if C is true whenever all A 1 , . . . , A n are true, i.e., every assignment of truth-values to the variables occurring in A 1 , . . . , A n , C which renders the formulae A 1 , . . . , A n true, renders the formula C true, too. If A 1 , . . . , A n | = C , we also say that C follows logically from A 1 , . . . , A n , Goranko

  7. Propositional logical consequence A propositional formula C is a logical consequence from the propositional formulae A 1 , . . . , A n , denoted A 1 , . . . , A n | = C , if C is true whenever all A 1 , . . . , A n are true, i.e., every assignment of truth-values to the variables occurring in A 1 , . . . , A n , C which renders the formulae A 1 , . . . , A n true, renders the formula C true, too. If A 1 , . . . , A n | = C , we also say that C follows logically from A 1 , . . . , A n , and that A 1 , . . . , A n logically imply C . Goranko

  8. Propositional logical consequence A propositional formula C is a logical consequence from the propositional formulae A 1 , . . . , A n , denoted A 1 , . . . , A n | = C , if C is true whenever all A 1 , . . . , A n are true, i.e., every assignment of truth-values to the variables occurring in A 1 , . . . , A n , C which renders the formulae A 1 , . . . , A n true, renders the formula C true, too. If A 1 , . . . , A n | = C , we also say that C follows logically from A 1 , . . . , A n , and that A 1 , . . . , A n logically imply C . Logical consequence is reducible to validity: A 1 , . . . , A n | = C iff A 1 ∧ . . . ∧ A n | = C iff | = ( A 1 ∧ . . . ∧ A n ) → C . Goranko

  9. Propositional logical consequence is reducible to validity Goranko

  10. Propositional logical consequence is reducible to validity Proposition For any propositional formulae A 1 , . . . , A n , B, the following are equivalent: Goranko

  11. Propositional logical consequence is reducible to validity Proposition For any propositional formulae A 1 , . . . , A n , B, the following are equivalent: 1. A 1 , . . . , A n | = B Goranko

  12. Propositional logical consequence is reducible to validity Proposition For any propositional formulae A 1 , . . . , A n , B, the following are equivalent: 1. A 1 , . . . , A n | = B 2. A 1 ∧ . . . ∧ A n | = B Goranko

  13. Propositional logical consequence is reducible to validity Proposition For any propositional formulae A 1 , . . . , A n , B, the following are equivalent: 1. A 1 , . . . , A n | = B 2. A 1 ∧ . . . ∧ A n | = B 3. | = ( A 1 ∧ . . . ∧ A n ) → B Goranko

  14. Propositional logical consequence is reducible to validity Proposition For any propositional formulae A 1 , . . . , A n , B, the following are equivalent: 1. A 1 , . . . , A n | = B 2. A 1 ∧ . . . ∧ A n | = B 3. | = ( A 1 ∧ . . . ∧ A n ) → B 4. | = A 1 → ( . . . → ( A n → B ) . . . ) Goranko

  15. Testing propositional consequence with truth tables Example 1 Goranko

  16. Testing propositional consequence with truth tables Example 1 ? p , p → q | = q Goranko

  17. Testing propositional consequence with truth tables Example 1 ? p , p → q | = q p q p p → q q Goranko

  18. Testing propositional consequence with truth tables Example 1 ? p , p → q | = q p q p p → q q T T T T T Goranko

  19. Testing propositional consequence with truth tables Example 1 ? p , p → q | = q p q p p → q q T T T T T T F T F F Goranko

  20. Testing propositional consequence with truth tables Example 1 ? p , p → q | = q p q p p → q q T T T T T T F T F F F T F T T Goranko

  21. Testing propositional consequence with truth tables Example 1 ? p , p → q | = q p q p p → q q T T T T T T F T F F F T F T T F F F T F Goranko

  22. Testing propositional consequence with truth tables Example 1 ? p , p → q | = q p q p p → q q T T T T T T F T F F F T F T T F F F T F Yes. Goranko

  23. Testing propositional consequence with truth tables Example 2 Goranko

  24. Testing propositional consequence with truth tables Example 2 ? p → q | = q → p Goranko

  25. Testing propositional consequence with truth tables Example 2 ? p → q | = q → p p → q q → p p q Goranko

  26. Testing propositional consequence with truth tables Example 2 ? p → q | = q → p p → q q → p p q T T T T Goranko

  27. Testing propositional consequence with truth tables Example 2 ? p → q | = q → p p → q q → p p q T T T T T F F T Goranko

  28. Testing propositional consequence with truth tables Example 2 ? p → q | = q → p p → q q → p p q T T T T T F F T F F T T Goranko

  29. Testing propositional consequence with truth tables Example 2 ? p → q | = q → p p → q q → p p q T T T T T F F T F F T T F F ... ... Goranko

  30. Testing propositional consequence with truth tables Example 2 ? p → q | = q → p p → q q → p p q T T T T T F F T F F T T F F ... ... No. Goranko

  31. Testing propositional consequence with truth tables Example 3 Goranko

  32. Testing propositional consequence with truth tables Example 3 ? p → r , q → r | = ( p ∨ q ) → r Goranko

  33. Testing propositional consequence with truth tables Example 3 ? p → r , q → r | = ( p ∨ q ) → r p q r p → r q → r p ∨ q ( p ∨ q ) → r Goranko

  34. Testing propositional consequence with truth tables Example 3 ? p → r , q → r | = ( p ∨ q ) → r p q r p → r q → r p ∨ q ( p ∨ q ) → r T T T T T T T Goranko

  35. Testing propositional consequence with truth tables Example 3 ? p → r , q → r | = ( p ∨ q ) → r p q r p → r q → r p ∨ q ( p ∨ q ) → r T T T T T T T T T F F F T F Goranko

  36. Testing propositional consequence with truth tables Example 3 ? p → r , q → r | = ( p ∨ q ) → r p q r p → r q → r p ∨ q ( p ∨ q ) → r T T T T T T T T T F F F T F T F T T T T T Goranko

  37. Testing propositional consequence with truth tables Example 3 ? p → r , q → r | = ( p ∨ q ) → r p q r p → r q → r p ∨ q ( p ∨ q ) → r T T T T T T T T T F F F T F T F T T T T T T F F F T T F Goranko

  38. Testing propositional consequence with truth tables Example 3 ? p → r , q → r | = ( p ∨ q ) → r p q r p → r q → r p ∨ q ( p ∨ q ) → r T T T T T T T T T F F F T F T F T T T T T T F F F T T F F T T T T T T Goranko

  39. Testing propositional consequence with truth tables Example 3 ? p → r , q → r | = ( p ∨ q ) → r p q r p → r q → r p ∨ q ( p ∨ q ) → r T T T T T T T T T F F F T F T F T T T T T T F F F T T F F T T T T T T F T F T F T F Goranko

  40. Testing propositional consequence with truth tables Example 3 ? p → r , q → r | = ( p ∨ q ) → r p q r p → r q → r p ∨ q ( p ∨ q ) → r T T T T T T T T T F F F T F T F T T T T T T F F F T T F F T T T T T T F T F T F T F F F T T T F T Goranko

  41. Testing propositional consequence with truth tables Example 3 ? p → r , q → r | = ( p ∨ q ) → r p q r p → r q → r p ∨ q ( p ∨ q ) → r T T T T T T T T T F F F T F T F T T T T T T F F F T T F F T T T T T T F T F T F T F F F T T T F T F F F T T F T Goranko

  42. Testing propositional consequence with truth tables Example 3 ? p → r , q → r | = ( p ∨ q ) → r p q r p → r q → r p ∨ q ( p ∨ q ) → r T T T T T T T T T F F F T F T F T T T T T T F F F T T F F T T T T T T F T F T F T F F F T T T F T F F F T T F T Yes. Goranko

  43. Sound rules of propositional inference Goranko

  44. Sound rules of propositional inference A rule of propositional inference (for short, inference rule) is a scheme: P 1 , . . . , P n , C Goranko

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