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# Indirect-table Analysis

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## Indirect-table AnalysisPresentation 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