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Indirect-table Analysis

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Janet Stemwedel
Janet Stemwedel
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Indirect-Table Analysis Phil 57 section 3 San Jose State University Fall 2010
What are truth-tables good for? ,[object Object],[object Object],[object Object]
Sometimes we don’t need a full truth-table! ,[object Object],[object Object],[object Object],[object Object]
Example: P  Q, Q / P
Example: P  Q, Q / P Pr1 Pr2 C P Q P  Q Q P
Example: P  Q, Q / P ,[object Object],Pr1 Pr2 C P Q P  Q Q P
Example: P  Q, Q / P ,[object Object],Pr1 Pr2 C P Q P  Q Q P F F
Example: P  Q, Q / P ,[object Object],[object Object],Pr1 Pr2 C P Q P  Q Q P F F
Example: P  Q, Q / P ,[object Object],[object Object],Pr1 Pr2 C P Q P  Q Q P F T T F
Example: P  Q, Q / P ,[object Object],[object Object],Pr1 Pr2 C P Q P  Q Q P F T T T F
Example: P  Q, Q / P ,[object Object],[object Object],[object Object],Pr1 Pr2 C P Q P  Q Q P F T T T F
Detailed strategy:
Detailed strategy: ,[object Object]
Detailed strategy: ,[object Object],[object Object]
Detailed strategy: ,[object Object],[object Object],[object Object]
Detailed strategy: ,[object Object],[object Object],[object Object],[object Object]
Example 2: P  Q, P / Q ,[object Object]
Example 2: P  Q, P / Q Pr1 Pr2 C P  Q P Q
Example 2: P  Q, P / Q ,[object Object],Pr1 Pr2 C P  Q P Q
Example 2: P  Q, P / Q Pr1 Pr2 C P  Q P Q F
Example 2: P  Q, P / Q ,[object Object],Pr1 Pr2 C P  Q P Q F
Example 2: P  Q, P / Q Pr1 Pr2 C P  Q P Q T F
Example 2: P  Q, P / Q Pr1 Pr2 C P  Q P Q T T F
Example 2: P  Q, P / Q Pr1 Pr2 C P  Q P Q T F T F
Example 2: P  Q, P / Q Pr1 Pr2 C P  Q P Q T F T F F
Example 2: P  Q, P / Q ,[object Object],Pr1 Pr2 C P  Q P Q T F T F F
Example 2: P  Q, P / Q ,[object Object],[object Object],Pr1 Pr2 C P  Q P Q T F T F F
Example 3: P  Q, (R  Q)  S / P
Example 3: P  Q, (R  Q)  S / P ,[object Object]
Example 3: P  Q, (R  Q)  S / P Pr1 Pr2 C P  Q (R  Q)  S P
Example 3: P  Q, (R  Q)  S / P ,[object Object],Pr1 Pr2 C P  Q (R  Q)  S P
Example 3: P  Q, (R  Q)  S / P Pr1 Pr2 C P  Q (R  Q)  S P F
Example 3: P  Q, (R  Q)  S / P ,[object Object],Pr1 Pr2 C P  Q (R  Q)  S P F
Example 3: P  Q, (R  Q)  S / P Pr1 Pr2 C P  Q (R  Q)  S P F F
Example 3: P  Q, (R  Q)  S / P Pr1 Pr2 C P  Q (R  Q)  S P F F T
Example 3: P  Q, (R  Q)  S / P Pr1 Pr2 C P  Q (R  Q)  S P F T F T
Example 3: P  Q, (R  Q)  S / P Pr1 Pr2 C P  Q (R  Q)  S P F T F T T
Example 3: P  Q, (R  Q)  S / P ,[object Object],Pr1 Pr2 C P  Q (R  Q)  S P F T F T T
Example 3: P  Q, (R  Q)  S / P ,[object Object],[object Object],Pr1 Pr2 C P  Q (R  Q)  S P F T F T T
Example 4: P  Q, Q  R, ~S  V / V  P ,[object Object]
Example 4: P  Q, Q  R, ~S  V / V  P Pr1 Pr2 Pr3 C P  Q Q  R ~S  V V  P
Example 4: P  Q, Q  R, ~S  V / V  P ,[object Object],Pr1 Pr2 Pr3 C P  Q Q  R ~S  V V  P
Example 4: P  Q, Q  R, ~S  V / V  P ,[object Object],[object Object],Pr1 Pr2 Pr3 C P  Q Q  R ~S  V V  P
Example 4: P  Q, Q  R, ~S  V / V  P Pr1 Pr2 Pr3 C P  Q Q  R ~S  V V  P F F
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
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
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
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
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
Example 4: P  Q, Q  R, ~S  V / V  P ,[object Object],Pr1 Pr2 Pr3 C P  Q Q  R ~S  V V  P F F F T F F F F T
Example 4: P  Q, Q  R, ~S  V / V  P ,[object Object],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
Example 4: P  Q, Q  R, ~S  V / V  P ,[object Object],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
Example 4: P  Q, Q  R, ~S  V / V  P ,[object Object],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
Example 4: P  Q, Q  R, ~S  V / V  P ,[object Object],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
Example 4: P  Q, Q  R, ~S  V / V  P ,[object Object],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
Example 4: P  Q, Q  R, ~S  V / V  P ,[object Object],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
Example 4: P  Q, Q  R, ~S  V / V  P ,[object Object],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
Example 4: P  Q, Q  R, ~S  V / V  P ,[object Object],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
Example 4: P  Q, Q  R, ~S  V / V  P ,[object Object],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
Example 4: P  Q, Q  R, ~S  V / V  P ,[object Object],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
Example 4: P  Q, Q  R, ~S  V / V  P ,[object Object],[object Object],[object Object],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
Example 4: P  Q, Q  R, ~S  V / V  P ,[object Object],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
Example 4: P  Q, Q  R, ~S  V / V  P ,[object Object],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
Example 4: P  Q, Q  R, ~S  V / V  P ,[object Object],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
Example 4: P  Q, Q  R, ~S  V / V  P ,[object Object],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
Example 4: P  Q, Q  R, ~S  V / V  P ,[object Object],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
Example 4: P  Q, Q  R, ~S  V / V  P ,[object Object],[object Object],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
Indirect-tables to determine if a set of formulae is satisfiable:
Indirect-tables to determine if a set of formulae is satisfiable: ,[object Object]
Indirect-tables to determine if a set of formulae is satisfiable: ,[object Object],[object Object]
Indirect-tables to determine if a set of formulae is satisfiable: ,[object Object],[object Object],[object Object]
Example: P  Q, ~P  Q P  Q ~P  Q
Example: P  Q, ~P  Q P  Q ~P  Q T T
Example: P  Q, ~P  Q P  Q ~P  Q T T F F T F
Example: P  Q, ~P  Q Satisfiable P  Q ~P  Q T T F F T F
Example: P  Q, P  ~Q P  Q P  ~Q
Example: P  Q, P  ~Q P  Q P  ~Q T T
Example: P  Q, P  ~Q P  Q P  ~Q T T T T F
Example: P  Q, P  ~Q P  Q P  ~Q T T T F T T F
Example: P  Q, P  ~Q P  Q P  ~Q T T T F T T F F
Example: P  Q, P  ~Q Unsatisfiable P  Q P  ~Q T T T F T T F F
Example: P  ~Q, P  Q P  ~Q P  Q T T T F T F F T F
Example: P  ~Q, P  Q P  ~Q P  Q T T T T T F T F F T F
Example: P  ~Q, P  Q P  ~Q P  Q T F T T T T F T F F T F
Example: P  ~Q, P  Q P  ~Q P  Q T F F T T T T F T F F T F
Example: P  ~Q, P  Q P  ~Q P  Q T F F T T T T T F F T F F T F
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
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
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
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
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
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
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
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.
Next time: Quiz #3 ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

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Indirect-table Analysis

  • 1. Indirect-Table Analysis Phil 57 section 3 San Jose State University Fall 2010
  • 2.
  • 3.
  • 4. Example: P  Q, Q / P
  • 5. Example: P  Q, Q / P Pr1 Pr2 C P Q P  Q Q P
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  • 17.
  • 18. Example 2: P  Q, P / Q Pr1 Pr2 C P  Q P Q
  • 19.
  • 20. Example 2: P  Q, P / Q Pr1 Pr2 C P  Q P Q F
  • 21.
  • 22. Example 2: P  Q, P / Q Pr1 Pr2 C P  Q P Q T F
  • 23. Example 2: P  Q, P / Q Pr1 Pr2 C P  Q P Q T T F
  • 24. Example 2: P  Q, P / Q Pr1 Pr2 C P  Q P Q T F T F
  • 25. Example 2: P  Q, P / Q Pr1 Pr2 C P  Q P Q T F T F F
  • 26.
  • 27.
  • 28. Example 3: P  Q, (R  Q)  S / P
  • 29.
  • 30. Example 3: P  Q, (R  Q)  S / P Pr1 Pr2 C P  Q (R  Q)  S P
  • 31.
  • 32. Example 3: P  Q, (R  Q)  S / P Pr1 Pr2 C P  Q (R  Q)  S P F
  • 33.
  • 34. Example 3: P  Q, (R  Q)  S / P Pr1 Pr2 C P  Q (R  Q)  S P F F
  • 35. Example 3: P  Q, (R  Q)  S / P Pr1 Pr2 C P  Q (R  Q)  S P F F T
  • 36. Example 3: P  Q, (R  Q)  S / P Pr1 Pr2 C P  Q (R  Q)  S P F T F T
  • 37. Example 3: P  Q, (R  Q)  S / P Pr1 Pr2 C P  Q (R  Q)  S P F T F T T
  • 38.
  • 39.
  • 40.
  • 41. Example 4: P  Q, Q  R, ~S  V / V  P Pr1 Pr2 Pr3 C P  Q Q  R ~S  V V  P
  • 42.
  • 43.
  • 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. 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. 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. 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. 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. 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
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  • 67.
  • 68. Indirect-tables to determine if a set of formulae is satisfiable:
  • 69.
  • 70.
  • 71.
  • 72. Example: P  Q, ~P  Q P  Q ~P  Q
  • 73. Example: P  Q, ~P  Q P  Q ~P  Q T T
  • 74. Example: P  Q, ~P  Q P  Q ~P  Q T T F F T F
  • 75. Example: P  Q, ~P  Q Satisfiable P  Q ~P  Q T T F F T F
  • 76. Example: P  Q, P  ~Q P  Q P  ~Q
  • 77. Example: P  Q, P  ~Q P  Q P  ~Q T T
  • 78. Example: P  Q, P  ~Q P  Q P  ~Q T T T T F
  • 79. Example: P  Q, P  ~Q P  Q P  ~Q T T T F T T F
  • 80. Example: P  Q, P  ~Q P  Q P  ~Q T T T F T T F F
  • 81. Example: P  Q, P  ~Q Unsatisfiable P  Q P  ~Q T T T F T T F F
  • 82. Example: P  ~Q, P  Q P  ~Q P  Q T T T F T F F T F
  • 83. Example: P  ~Q, P  Q P  ~Q P  Q T T T T T F T F F T F
  • 84. Example: P  ~Q, P  Q P  ~Q P  Q T F T T T T F T F F T F
  • 85. Example: P  ~Q, P  Q P  ~Q P  Q T F F T T T T F T F F T F
  • 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. 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. 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. 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. 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. 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. 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. 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. 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.