Indirect-table Analysis

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Indirect-table Analysis

  1. 1. Indirect-Table Analysis Phil 57 section 3 San Jose State University Fall 2010
  2. 2. What are truth-tables good for? <ul><li>Determining the logical status of a single proposition. </li></ul><ul><li>Determining the logical status of a group of propositions. </li></ul><ul><li>Determining the validity of an argument. </li></ul>
  3. 3. Sometimes we don’t need a full truth-table! <ul><li>Invalid argument has true premises and false conclusion . </li></ul><ul><li>Strategy: </li></ul><ul><li>Find rows that make conclusion F. </li></ul><ul><li>Find rows that make premises T. </li></ul>
  4. 4. Example: P  Q, Q / P
  5. 5. Example: P  Q, Q / P Pr1 Pr2 C P Q P  Q Q P
  6. 6. Example: P  Q, Q / P <ul><li>Conclusion is P ( only need rows where P is F ) </li></ul>Pr1 Pr2 C P Q P  Q Q P
  7. 7. Example: P  Q, Q / P <ul><li>Conclusion is P ( only need rows where P is F ) </li></ul>Pr1 Pr2 C P Q P  Q Q P F F
  8. 8. Example: P  Q, Q / P <ul><li>Conclusion is P ( only need rows where P is F ) </li></ul><ul><li>Q is a premise ( only need row where Q is T ) </li></ul>Pr1 Pr2 C P Q P  Q Q P F F
  9. 9. Example: P  Q, Q / P <ul><li>Conclusion is P ( only need rows where P is F ) </li></ul><ul><li>Q is a premise ( only need row where Q is T ) </li></ul>Pr1 Pr2 C P Q P  Q Q P F T T F
  10. 10. Example: P  Q, Q / P <ul><li>Conclusion is P ( only need rows where P is F ) </li></ul><ul><li>Q is a premise ( only need row where Q is T ) </li></ul>Pr1 Pr2 C P Q P  Q Q P F T T T F
  11. 11. Example: P  Q, Q / P <ul><li>Conclusion is P ( only need rows where P is F ) </li></ul><ul><li>Q is a premise ( only need row where Q is T ) </li></ul><ul><li>Argument is INVALID. </li></ul>Pr1 Pr2 C P Q P  Q Q P F T T T F
  12. 12. Detailed strategy:
  13. 13. Detailed strategy: <ul><li>Write argument (premises and conclusion) at top of table columns. </li></ul>
  14. 14. Detailed strategy: <ul><li>Write argument (premises and conclusion) at top of table columns. </li></ul><ul><li>Make the conclusion false. </li></ul>
  15. 15. Detailed strategy: <ul><li>Write argument (premises and conclusion) at top of table columns. </li></ul><ul><li>Make the conclusion false. </li></ul><ul><li>Try to make the premises true without being forced to assign both T and F to any single atomic statement or formula. </li></ul>
  16. 16. Detailed strategy: <ul><li>Write argument (premises and conclusion) at top of table columns. </li></ul><ul><li>Make the conclusion false. </li></ul><ul><li>Try to make the premises true without being forced to assign both T and F to any single atomic statement or formula. </li></ul><ul><li>If forced to assign T and F to the same atomic statement or formula, the argument is valid. </li></ul>
  17. 17. Example 2: P  Q, P / Q <ul><li>Write argument (premises and conclusion) at top of table columns. </li></ul>
  18. 18. Example 2: P  Q, P / Q Pr1 Pr2 C P  Q P Q
  19. 19. Example 2: P  Q, P / Q <ul><li>2. Make the conclusion false. </li></ul>Pr1 Pr2 C P  Q P Q
  20. 20. Example 2: P  Q, P / Q Pr1 Pr2 C P  Q P Q F
  21. 21. Example 2: P  Q, P / Q <ul><li>3. Try to make the premises true without being forced to assign both T and F to any single atomic statement or formula. </li></ul>Pr1 Pr2 C P  Q P Q F
  22. 22. Example 2: P  Q, P / Q Pr1 Pr2 C P  Q P Q T F
  23. 23. Example 2: P  Q, P / Q Pr1 Pr2 C P  Q P Q T T F
  24. 24. Example 2: P  Q, P / Q Pr1 Pr2 C P  Q P Q T F T F
  25. 25. Example 2: P  Q, P / Q Pr1 Pr2 C P  Q P Q T F T F F
  26. 26. Example 2: P  Q, P / Q <ul><li>Can’t make conclusion F and both premises T. </li></ul>Pr1 Pr2 C P  Q P Q T F T F F
  27. 27. Example 2: P  Q, P / Q <ul><li>Can’t make conclusion F and both premises T. </li></ul><ul><li>Argument is valid! </li></ul>Pr1 Pr2 C P  Q P Q T F T F F
  28. 28. Example 3: P  Q, (R  Q)  S / P
  29. 29. Example 3: P  Q, (R  Q)  S / P <ul><li>Write argument (premises and conclusion) at top of table columns. </li></ul>
  30. 30. Example 3: P  Q, (R  Q)  S / P Pr1 Pr2 C P  Q (R  Q)  S P
  31. 31. Example 3: P  Q, (R  Q)  S / P <ul><li>2. Make the conclusion false. </li></ul>Pr1 Pr2 C P  Q (R  Q)  S P
  32. 32. Example 3: P  Q, (R  Q)  S / P Pr1 Pr2 C P  Q (R  Q)  S P F
  33. 33. Example 3: P  Q, (R  Q)  S / P <ul><li>3. Try to make the premises true without being forced to assign both T and F to any single atomic statement or formula. </li></ul>Pr1 Pr2 C P  Q (R  Q)  S P F
  34. 34. Example 3: P  Q, (R  Q)  S / P Pr1 Pr2 C P  Q (R  Q)  S P F F
  35. 35. Example 3: P  Q, (R  Q)  S / P Pr1 Pr2 C P  Q (R  Q)  S P F F T
  36. 36. Example 3: P  Q, (R  Q)  S / P Pr1 Pr2 C P  Q (R  Q)  S P F T F T
  37. 37. Example 3: P  Q, (R  Q)  S / P Pr1 Pr2 C P  Q (R  Q)  S P F T F T T
  38. 38. Example 3: P  Q, (R  Q)  S / P <ul><li>Made conclusion F and both premises T. </li></ul>Pr1 Pr2 C P  Q (R  Q)  S P F T F T T
  39. 39. Example 3: P  Q, (R  Q)  S / P <ul><li>Made conclusion F and both premises T. </li></ul><ul><li>Argument is invalid! </li></ul>Pr1 Pr2 C P  Q (R  Q)  S P F T F T T
  40. 40. Example 4: P  Q, Q  R, ~S  V / V  P <ul><li>Write argument (premises and conclusion) at top of table columns. </li></ul>
  41. 41. Example 4: P  Q, Q  R, ~S  V / V  P Pr1 Pr2 Pr3 C P  Q Q  R ~S  V V  P
  42. 42. Example 4: P  Q, Q  R, ~S  V / V  P <ul><li>2. Make the conclusion false. </li></ul>Pr1 Pr2 Pr3 C P  Q Q  R ~S  V V  P
  43. 43. Example 4: P  Q, Q  R, ~S  V / V  P <ul><li>2. Make the conclusion false. </li></ul><ul><li>Three different ways to make the disjunction. </li></ul>Pr1 Pr2 Pr3 C P  Q Q  R ~S  V V  P
  44. 44. Example 4: P  Q, Q  R, ~S  V / V  P Pr1 Pr2 Pr3 C P  Q Q  R ~S  V V  P F F
  45. 45. Example 4: P  Q, Q  R, ~S  V / V  P Pr1 Pr2 Pr3 C P  Q Q  R ~S  V V  P F F F
  46. 46. Example 4: P  Q, Q  R, ~S  V / V  P Pr1 Pr2 Pr3 C P  Q Q  R ~S  V V  P F F F T F
  47. 47. Example 4: P  Q, Q  R, ~S  V / V  P Pr1 Pr2 Pr3 C P  Q Q  R ~S  V V  P F F F T F F
  48. 48. Example 4: P  Q, Q  R, ~S  V / V  P Pr1 Pr2 Pr3 C P  Q Q  R ~S  V V  P F F F T F F F T
  49. 49. Example 4: P  Q, Q  R, ~S  V / V  P Pr1 Pr2 Pr3 C P  Q Q  R ~S  V V  P F F F T F F F F T
  50. 50. Example 4: P  Q, Q  R, ~S  V / V  P <ul><li>3. Try to make the premises true without being forced to assign both T and F to any single atomic statement or formula. </li></ul>Pr1 Pr2 Pr3 C P  Q Q  R ~S  V V  P F F F T F F F F T
  51. 51. Example 4: P  Q, Q  R, ~S  V / V  P <ul><li>Fill in values of P and V from each row. </li></ul>Pr1 Pr2 Pr3 C P  Q Q  R ~S  V V  P F T F F F F T T F F T T T F F T
  52. 52. Example 4: P  Q, Q  R, ~S  V / V  P <ul><li>Fill in values of P and V from each row. </li></ul>Pr1 Pr2 Pr3 C P  Q Q  R ~S  V V  P F F F F F F T T F F T F F F T
  53. 53. Example 4: P  Q, Q  R, ~S  V / V  P <ul><li>Work out column for Pr3. </li></ul>Pr1 Pr2 Pr3 C P  Q Q  R ~S  V V  P F F F F F F T T F F T F F F T
  54. 54. Example 4: P  Q, Q  R, ~S  V / V  P <ul><li>Work out column for Pr3. </li></ul>Pr1 Pr2 Pr3 C P  Q Q  R ~S  V V  P F T F F F F F T T T F F T T F F F T
  55. 55. Example 4: P  Q, Q  R, ~S  V / V  P <ul><li>Work out column for Pr3. </li></ul>Pr1 Pr2 Pr3 C P  Q Q  R ~S  V V  P F T T F F F F F T T T F F T T T F F F T
  56. 56. Example 4: P  Q, Q  R, ~S  V / V  P <ul><li>Work out column for Pr3. </li></ul>Pr1 Pr2 Pr3 C P  Q Q  R ~S  V V  P F T F T F F F F F T T T F F T T F T F F F T
  57. 57. Example 4: P  Q, Q  R, ~S  V / V  P <ul><li>Work out column for Pr3. </li></ul>Pr1 Pr2 Pr3 C P  Q Q  R ~S  V V  P F T F F T T F T F F F F F T F F T T T T F F T T T T T T T F T F F F T
  58. 58. Example 4: P  Q, Q  R, ~S  V / V  P <ul><li>Work out column for Pr1. </li></ul>Pr1 Pr2 Pr3 C P  Q Q  R ~S  V V  P F T F T F F F F F T T T F F T T F T F F F T
  59. 59. Example 4: P  Q, Q  R, ~S  V / V  P <ul><li>Work out column for Pr1. </li></ul>Pr1 Pr2 Pr3 C P  Q Q  R ~S  V V  P F T T F T F F F F F T T T T F F T T T F T F F F T
  60. 60. Example 4: P  Q, Q  R, ~S  V / V  P <ul><li>Work out column for Pr1. </li></ul>Pr1 Pr2 Pr3 C P  Q Q  R ~S  V V  P F T T F T F F F F F T T T T F F T T T T F T F F F T
  61. 61. Example 4: P  Q, Q  R, ~S  V / V  P <ul><li>Work out column for Pr1. </li></ul><ul><li>When P is F, Q could be T or F. </li></ul><ul><li>(Making Q false automatically makes Pr2 true) </li></ul>Pr1 Pr2 Pr3 C P  Q Q  R ~S  V V  P F T T F T F F F F F T T T T F F T T T T F T F F F T
  62. 62. Example 4: P  Q, Q  R, ~S  V / V  P <ul><li>Work out column for Pr2. </li></ul>Pr1 Pr2 Pr3 C P  Q Q  R ~S  V V  P F T F T F T F F F F F T F T T T F F T T T T F T F F F T
  63. 63. Example 4: P  Q, Q  R, ~S  V / V  P <ul><li>Work out column for Pr2. </li></ul>Pr1 Pr2 Pr3 C P  Q Q  R ~S  V V  P F T F F T F T F F F F F T F F T T T F F T T T T T F T F F F T
  64. 64. Example 4: P  Q, Q  R, ~S  V / V  P <ul><li>Work out column for Pr2. </li></ul>Pr1 Pr2 Pr3 C P  Q Q  R ~S  V V  P F T F F T T F T F F F F F T F F T T T T F F T T T T T T F T F F F T
  65. 65. Example 4: P  Q, Q  R, ~S  V / V  P <ul><li>Work out column for Pr2. </li></ul>Pr1 Pr2 Pr3 C P  Q Q  R ~S  V V  P F T F F T T F T F F F F F T F F T T T T F F T T T T T T T F T F F F T
  66. 66. Example 4: P  Q, Q  R, ~S  V / V  P <ul><li>Made conclusion F and all premises T. </li></ul>Pr1 Pr2 Pr3 C P  Q Q  R ~S  V V  P F T F F T T F T F F F F F T F F T T T T F F T T T T T T T F T F F F T
  67. 67. Example 4: P  Q, Q  R, ~S  V / V  P <ul><li>Made conclusion F and all premises T. </li></ul><ul><li>Argument is invalid! </li></ul>Pr1 Pr2 Pr3 C P  Q Q  R ~S  V V  P F T F F T T F T F F F F F T F F T T T T F F T T T T T T T F T F F F T
  68. 68. Indirect-tables to determine if a set of formulae is satisfiable:
  69. 69. Indirect-tables to determine if a set of formulae is satisfiable: <ul><li>Write formulae at top of table columns. </li></ul>
  70. 70. Indirect-tables to determine if a set of formulae is satisfiable: <ul><li>Write formulae at top of table columns. </li></ul><ul><li>Put a T under the main connective of each formula. </li></ul>
  71. 71. Indirect-tables to determine if a set of formulae is satisfiable: <ul><li>Write formulae at top of table columns. </li></ul><ul><li>Put a T under the main connective of each formula. </li></ul><ul><li>Try to find a distribution of truth-values that maintains the truth of the formulae. </li></ul>
  72. 72. Example: P  Q, ~P  Q P  Q ~P  Q
  73. 73. Example: P  Q, ~P  Q P  Q ~P  Q T T
  74. 74. Example: P  Q, ~P  Q P  Q ~P  Q T T F F T F
  75. 75. Example: P  Q, ~P  Q Satisfiable P  Q ~P  Q T T F F T F
  76. 76. Example: P  Q, P  ~Q P  Q P  ~Q
  77. 77. Example: P  Q, P  ~Q P  Q P  ~Q T T
  78. 78. Example: P  Q, P  ~Q P  Q P  ~Q T T T T F
  79. 79. Example: P  Q, P  ~Q P  Q P  ~Q T T T F T T F
  80. 80. Example: P  Q, P  ~Q P  Q P  ~Q T T T F T T F F
  81. 81. Example: P  Q, P  ~Q Unsatisfiable P  Q P  ~Q T T T F T T F F
  82. 82. Example: P  ~Q, P  Q P  ~Q P  Q T T T F T F F T F
  83. 83. Example: P  ~Q, P  Q P  ~Q P  Q T T T T T F T F F T F
  84. 84. Example: P  ~Q, P  Q P  ~Q P  Q T F T T T T F T F F T F
  85. 85. Example: P  ~Q, P  Q P  ~Q P  Q T F F T T T T F T F F T F
  86. 86. Example: P  ~Q, P  Q P  ~Q P  Q T F F T T T T T F F T F F T F
  87. 87. Example: P  ~Q, P  Q P  ~Q P  Q T F F T T T T T T F F T F F T F
  88. 88. Example: P  ~Q, P  Q P  ~Q P  Q T F F T T T T T T T F F T F F T F
  89. 89. Example: P  ~Q, P  Q P  ~Q P  Q T F F T T T T T T T F F T F F T T F F T F
  90. 90. Example: P  ~Q, P  Q P  ~Q P  Q T F F T T T T T T T F F T F F F F T F
  91. 91. Example: P  ~Q, P  Q P  ~Q P  Q T F F T T T T T T T F F T F F T F F T F
  92. 92. Example: P  ~Q, P  Q P  ~Q P  Q T F F T T T T T T T F F T F F T T F F T F
  93. 93. Example: P  ~Q, P  Q Satisfiable P  ~Q P  Q T F F T T T T T T T F F T F F T T F F T F
  94. 94. Summary of indirect-table tests: Test Procedure Results Satisfiability Place T under the main connective of each formula If there is at least one row where every formula can be T, the set is satisfiable. Validity Place T under every premise, F under conclusion If there is a row where premises are true and conclusion is false, argument is invalid.
  95. 95. Next time: Quiz #3 <ul><li>Translation from English to PL </li></ul><ul><li>Translation from PL to English </li></ul><ul><li>Truth-tables </li></ul><ul><li>To prepare: </li></ul><ul><li>HW #7 </li></ul><ul><li>HW #8 </li></ul>

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