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Indirect-Table Analysis Phil 57 section 3 San Jose State University Fall 2010
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Indirect-Table Analysis
Phil 57 section 3San 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: PQ, Q / P
Example: PQ, Q / P
Pr1 Pr2 CP Q PQ Q P
Example: PQ, Q / P
Conclusion is P (only need rows where P is F)
Pr1 Pr2 CP Q PQ Q P
Example: PQ, Q / P
Conclusion is P (only need rows where P is F)
Pr1 Pr2 CP Q PQ Q PF F
Example: PQ, 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 CP Q PQ Q PF F
Example: PQ, 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 CP Q PQ Q PF T T F
Example: PQ, 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 CP Q PQ Q PF T T T F
Example: PQ, 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 CP Q PQ Q PF T T T F
Detailed strategy:
Detailed strategy:
1. Write argument (premises and conclusion) at top of table columns.
Detailed strategy:
1. Write argument (premises and conclusion) at top of table columns.
2. Make the conclusion false.
Detailed strategy:
1. Write argument (premises and conclusion) at top of table columns.
2. Make the conclusion false.3. Try to make the premises true without being
forced to assign both T and F to any single atomic statement or formula.
Detailed strategy:
1. Write argument (premises and conclusion) at top of table columns.
2. Make the conclusion false.3. Try to make the premises true without being
forced to assign both T and F to any single atomic statement or formula.
4. If forced to assign T and F to the same atomic statement or formula, the argument is valid.
Example 2: PQ, P / Q
1. Write argument (premises and conclusion) at top of table columns.
Example 2: PQ, P / Q
Pr1 Pr2 CPQ P Q
Example 2: PQ, P / Q
2. Make the conclusion false.
Pr1 Pr2 CPQ P Q
Example 2: PQ, P / Q
Pr1 Pr2 CPQ P Q
F
Example 2: PQ, 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 CPQ P Q
F
Example 2: PQ, P / Q
Pr1 Pr2 CPQ P Q
T F
Example 2: PQ, P / Q
Pr1 Pr2 CPQ P Q
T T F
Example 2: PQ, P / Q
Pr1 Pr2 CPQ P QT F T F
Example 2: PQ, P / Q
Pr1 Pr2 CPQ P QT F T F
F
Example 2: PQ, P / Q
Can’t make conclusion F and both premises T.
Pr1 Pr2 CPQ P QT F T F
F
Example 2: PQ, P / Q
Can’t make conclusion F and both premises T.Argument is valid!
Pr1 Pr2 CPQ P QT F T F
F
Example 3: PQ, (RQ)S / P
Example 3: PQ, (RQ)S / P
1. Write argument (premises and conclusion) at top of table columns.
Example 3: PQ, (RQ)S / P
Pr1 Pr2 CPQ (RQ) S P
Example 3: PQ, (RQ)S / P
2. Make the conclusion false.
Pr1 Pr2 CPQ (RQ) S P
Example 3: PQ, (RQ)S / P
Pr1 Pr2 CPQ (RQ) S P
F
Example 3: PQ, (RQ)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 CPQ (RQ) S P
F
Example 3: PQ, (RQ)S / P
Pr1 Pr2 CPQ (RQ) S P
F F
Example 3: PQ, (RQ)S / P
Pr1 Pr2 CPQ (RQ) S P
F FT
Example 3: PQ, (RQ)S / P
Pr1 Pr2 CPQ (RQ) S P
F T FT
Example 3: PQ, (RQ)S / P
Pr1 Pr2 CPQ (RQ) S P
F T FT T
Example 3: PQ, (RQ)S / P
Made conclusion F and both premises T.
Pr1 Pr2 CPQ (RQ) S P
F T FT T
Example 3: PQ, (RQ)S / P
Made conclusion F and both premises T.Argument is invalid!
Pr1 Pr2 CPQ (RQ) S P
F T FT T
Example 4: PQ, QR, ~S V / VP
1. Write argument (premises and conclusion) at top of table columns.
Example 4: PQ, QR, ~S V / VP
Pr1 Pr2 Pr3 CPQ QR ~S V V P
Example 4: PQ, QR, ~S V / VP
2. Make the conclusion false.
Pr1 Pr2 Pr3 CPQ QR ~S V V P
Example 4: PQ, QR, ~S V / VP
2. Make the conclusion false.Three different ways to make the disjunction.
Pr1 Pr2 Pr3 CPQ QR ~S V V P
Example 4: PQ, QR, ~S V / VP
Pr1 Pr2 Pr3 CPQ QR ~S V V P
F F
Example 4: PQ, QR, ~S V / VP
Pr1 Pr2 Pr3 CPQ QR ~S V V P
F F F
Example 4: PQ, QR, ~S V / VP
Pr1 Pr2 Pr3 CPQ QR ~S V V P
F F F T F
Example 4: PQ, QR, ~S V / VP
Pr1 Pr2 Pr3 CPQ QR ~S V V P
F F F T F F
Example 4: PQ, QR, ~S V / VP
Pr1 Pr2 Pr3 CPQ QR ~S V V P
F F F T F F F T
Example 4: PQ, QR, ~S V / VP
Pr1 Pr2 Pr3 CPQ QR ~S V V P
F F F T F F F F T
Example 4: PQ, QR, ~S V / VP
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 CPQ QR ~S V V P
F F F T F F F F T
Example 4: PQ, QR, ~S V / VP
Fill in values of P and V from each row.
Pr1 Pr2 Pr3 CPQ QR ~S V V P
F T
F F F
F T
T F F
T T T F F T
Example 4: PQ, QR, ~S V / VP
Fill in values of P and V from each row.
Pr1 Pr2 Pr3 CPQ QR ~S V V P
F F F F F F T
T F F
T F F F T
Example 4: PQ, QR, ~S V / VP
Work out column for Pr3.
Pr1 Pr2 Pr3 CPQ QR ~S V V P
F F F F F F T
T F F
T F F F T
Example 4: PQ, QR, ~S V / VP
Work out column for Pr3.
Pr1 Pr2 Pr3 CPQ QR ~S V V P
F T F F F F F T T
T F F
T T F F F T
Example 4: PQ, QR, ~S V / VP
Work out column for Pr3.
Pr1 Pr2 Pr3 CPQ QR ~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: PQ, QR, ~S V / VP
Work out column for Pr3.
Pr1 Pr2 Pr3 CPQ QR ~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: PQ, QR, ~S V / VP
Work out column for Pr3.
Pr1 Pr2 Pr3 CPQ QR ~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: PQ, QR, ~S V / VP
Work out column for Pr1.
Pr1 Pr2 Pr3 CPQ QR ~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: PQ, QR, ~S V / VP
Work out column for Pr1.
Pr1 Pr2 Pr3 CPQ QR ~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: PQ, QR, ~S V / VP
Work out column for Pr1.
Pr1 Pr2 Pr3 CPQ QR ~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: PQ, QR, ~S V / VP
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 CPQ QR ~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: PQ, QR, ~S V / VP
Work out column for Pr2.
Pr1 Pr2 Pr3 CPQ QR ~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: PQ, QR, ~S V / VP
Work out column for Pr2.
Pr1 Pr2 Pr3 CPQ QR ~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: PQ, QR, ~S V / VP
Work out column for Pr2.
Pr1 Pr2 Pr3 CPQ QR ~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: PQ, QR, ~S V / VP
Work out column for Pr2.
Pr1 Pr2 Pr3 CPQ QR ~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: PQ, QR, ~S V / VP
Made conclusion F and all premises T.
Pr1 Pr2 Pr3 CPQ QR ~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: PQ, QR, ~S V / VP
Made conclusion F and all premises T.Argument is invalid!
Pr1 Pr2 Pr3 CPQ QR ~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:
1. Write formulae at top of table columns.
Indirect-tables to determine if a set of formulae is satisfiable:
1. Write formulae at top of table columns.2. Put a T under the main connective of each
formula.
Indirect-tables to determine if a set of formulae is satisfiable:
1. Write formulae at top of table columns.2. Put a T under the main connective of each
formula.3. Try to find a distribution of truth-values that
maintains the truth of the formulae.
Example: PQ, ~PQ
PQ ~PQ
Example: PQ, ~PQ
PQ ~PQT T
Example: PQ, ~PQ
PQ ~PQT T
F F T F
Example: PQ, ~PQ
PQ ~PQT T
F F T F
Satisfiable
Example: PQ, P~Q
PQ P ~Q
Example: PQ, P~Q
PQ P ~QT T
Example: PQ, P~Q
PQ P ~QT T
T T F
Example: PQ, P~Q
PQ P ~QT T
T F T T F
Example: PQ, P~Q
PQ P ~QT T
T F T T FF
Example: PQ, P~Q
PQ P ~QT T
T F T T FF
Unsatisfiable
Example: P~Q, PQ
P~Q P QT T TF T FF T F
Example: P~Q, PQ
P~Q P QT T T T T
F T FF T F
Example: P~Q, PQ
P~Q P QT F T T T T
F T FF T F
Example: P~Q, PQ
P~Q P QT F F T T T T
F T FF T F
Example: P~Q, PQ
P~Q P QT F F T T T TT F F T F
F T F
Example: P~Q, PQ
P~Q P QT F F T T T TT T F F T F
F T F
Example: P~Q, PQ
P~Q P QT F F T T T TT T T F F T F
F T F
Example: P~Q, PQ
P~Q P QT F F T T T TT T T F F T FF T T F F T F
Example: P~Q, PQ
P~Q P QT F F T T T TT T T F F T FF F F T F
Example: P~Q, PQ
P~Q P QT F F T T T TT T T F F T FF T F F T F
Example: P~Q, PQ
P~Q P QT F F T T T TT T T F F T FF T T F F T F
Example: P~Q, PQ
P~Q P QT F F T T T TT T T F F T FF T T F F T F
Satisfiable
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-tablesTo prepare:HW #7HW #8
