Indirect-table Analysis
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Indirect-table Analysis

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Indirect-table Analysis Indirect-table Analysis Presentation Transcript

  • Indirect-Table Analysis Phil 57 section 3 San Jose State University Fall 2010
  • What are truth-tables good for?
    • Determining the logical status of a single proposition.
    • Determining the logical status of a group of propositions.
    • Determining the validity of an argument.
  • Sometimes we don’t need a full truth-table!
    • Invalid argument has true premises and false conclusion .
    • Strategy:
    • Find rows that make conclusion F.
    • Find rows that make premises T.
  • Example: P  Q, Q / P
  • Example: P  Q, Q / P Pr1 Pr2 C P Q P  Q Q P
  • Example: P  Q, Q / P
    • Conclusion is P ( only need rows where P is F )
    Pr1 Pr2 C P Q P  Q Q P
  • Example: P  Q, Q / P
    • Conclusion is P ( only need rows where P is F )
    Pr1 Pr2 C P Q P  Q Q P F F
  • Example: P  Q, Q / P
    • Conclusion is P ( only need rows where P is F )
    • Q is a premise ( only need row where Q is T )
    Pr1 Pr2 C P Q P  Q Q P F F
  • Example: P  Q, Q / P
    • Conclusion is P ( only need rows where P is F )
    • Q is a premise ( only need row where Q is T )
    Pr1 Pr2 C P Q P  Q Q P F T T F
  • Example: P  Q, Q / P
    • Conclusion is P ( only need rows where P is F )
    • Q is a premise ( only need row where Q is T )
    Pr1 Pr2 C P Q P  Q Q P F T T T F
  • Example: P  Q, Q / P
    • Conclusion is P ( only need rows where P is F )
    • Q is a premise ( only need row where Q is T )
    • Argument is INVALID.
    Pr1 Pr2 C P Q P  Q Q P F T T T F
  • Detailed strategy:
  • Detailed strategy:
    • Write argument (premises and conclusion) at top of table columns.
  • Detailed strategy:
    • Write argument (premises and conclusion) at top of table columns.
    • Make the conclusion false.
  • Detailed strategy:
    • Write argument (premises and conclusion) at top of table columns.
    • Make the conclusion false.
    • Try to make the premises true without being forced to assign both T and F to any single atomic statement or formula.
  • Detailed strategy:
    • Write argument (premises and conclusion) at top of table columns.
    • Make the conclusion false.
    • Try to make the premises true without being forced to assign both T and F to any single atomic statement or formula.
    • If forced to assign T and F to the same atomic statement or formula, the argument is valid.
  • Example 2: P  Q, P / Q
    • Write argument (premises and conclusion) at top of table columns.
  • Example 2: P  Q, P / Q Pr1 Pr2 C P  Q P Q
  • Example 2: P  Q, P / Q
    • 2. Make the conclusion false.
    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
    • 3. Try to make the premises true without being forced to assign both T and F to any single atomic statement or formula.
    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
    • Can’t make conclusion F and both premises T.
    Pr1 Pr2 C P  Q P Q T F T F F
  • Example 2: P  Q, P / Q
    • Can’t make conclusion F and both premises T.
    • Argument is valid!
    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
    • Write argument (premises and conclusion) at top of table columns.
  • 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
    • 2. Make the conclusion false.
    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
    • 3. Try to make the premises true without being forced to assign both T and F to any single atomic statement or formula.
    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
    • Made conclusion F and both premises T.
    Pr1 Pr2 C P  Q (R  Q)  S P F T F T T
  • Example 3: P  Q, (R  Q)  S / P
    • Made conclusion F and both premises T.
    • Argument is invalid!
    Pr1 Pr2 C P  Q (R  Q)  S P F T F T T
  • Example 4: P  Q, Q  R, ~S  V / V  P
    • Write argument (premises and conclusion) at top of table columns.
  • 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
    • 2. Make the conclusion false.
    Pr1 Pr2 Pr3 C P  Q Q  R ~S  V V  P
  • Example 4: P  Q, Q  R, ~S  V / V  P
    • 2. Make the conclusion false.
    • Three different ways to make the disjunction.
    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
    • 3. Try to make the premises true without being forced to assign both T and F to any single atomic statement or formula.
    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
    • Fill in values of P and V from each row.
    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
    • Fill in values of P and V from each row.
    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
    • Work out column for Pr3.
    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
    • Work out column for Pr3.
    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
    • Work out column for Pr3.
    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
    • Work out column for Pr3.
    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
    • Work out column for Pr3.
    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
    • Work out column for Pr1.
    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
    • Work out column for Pr1.
    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
    • Work out column for Pr1.
    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
    • Work out column for Pr1.
    • When P is F, Q could be T or F.
    • (Making Q false automatically makes Pr2 true)
    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
    • Work out column for Pr2.
    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
    • Work out column for Pr2.
    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
    • Work out column for Pr2.
    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
    • Work out column for Pr2.
    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
    • Made conclusion F and all premises T.
    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
    • Made conclusion F and all premises T.
    • Argument is invalid!
    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:
    • Write formulae at top of table columns.
  • Indirect-tables to determine if a set of formulae is satisfiable:
    • Write formulae at top of table columns.
    • Put a T under the main connective of each formula.
  • Indirect-tables to determine if a set of formulae is satisfiable:
    • Write formulae at top of table columns.
    • Put a T under the main connective of each formula.
    • Try to find a distribution of truth-values that maintains the truth of the formulae.
  • 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
    • Translation from English to PL
    • Translation from PL to English
    • Truth-tables
    • To prepare:
    • HW #7
    • HW #8