Truth table analysis
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![Page 1: Truth table analysis](https://reader038.fdocuments.in/reader038/viewer/2022103004/55d505ddbb61eb4b338b4606/html5/thumbnails/1.jpg)
Truth-Table Analysis
Phil 57 section 3San Jose State University
Fall 2010
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What are truth-tables good for?
• Working out the truth value of a formula from the truth values of the atomic formulae in it.
• Defining the behavior of logical connectives used to build complex formulae.
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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.
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The logical status of a proposition.
Some propositions are always true:It will rain or it won’t.
Some propositions are never true:The universe is empty and not empty.
Most propositions are true in some conditions and false in others:Today is hot.
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The logical status of a proposition.
Can use truth-tables to distinguish these different kinds of propositions.
Logical status Description Truth-Table
Tautology Always true Every row T
Contradiction Always false Every row F
Contingent Depends on circumstances
Some row T, some row F
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The logical status of a proposition.
Consider P~P
P ~P P~P
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The logical status of a proposition.
Consider P~P
P ~P P~P
T
F
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The logical status of a proposition.
Consider P~P
P ~P P~P
T F
F T
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The logical status of a proposition.
Consider P~P
P ~P P~P
T F F
F T F
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The logical status of a proposition.
Consider P~P
Truth-value for P~P is F in every row. (Contradiction)
P ~P P~P
T F F
F T F
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The logical status of a proposition.
Consider P~P
P ~P P~P
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The logical status of a proposition.
Consider P~P
P ~P P~P
T
F
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The logical status of a proposition.
Consider P~P
P ~P P~P
T F
F T
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The logical status of a proposition.
Consider P~P
P ~P P~P
T F T
F T T
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The logical status of a proposition.
Consider P~P
Truth-value for P~P is T in every row. (Tautology)
P ~P P~P
T F T
F T T
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The logical status of a group of propositions.
• Do two different claims mean the same thing?• Can two different claims both be true?• Must one of a pair of claims be false if the
other is true?
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The logical status of a group of propositions.
Logical Status Description Truth-Table
Equivalent Mean the same thing, logically
The truth-tables are identical
Satisfiable/consistent
Possibly all true. There is some row where all propositions are true.
Unsatisfiable/inconsistent
Can’t all be true. No row where all propositions are true.
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Rules for building truth-tables:As many columns as:• Statement letters• Non-atomic formulae• The formula to be computed
2N rows, where N= number of atomic statement letters.
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Truth-table to evaluate ((PQ)~R) 1 2 3 4 5 6 (formula)
P Q R ~R (PQ) ((PQ)~R)
1
2
3
4
5
6
7
8
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Truth-table to evaluate ((PQ)~R) 1 2 3 4 5 6 (formula)
P Q R ~R (PQ) ((PQ)~R)
1 T2 T3 T4 T5 F6 F7 F8 F
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Truth-table to evaluate ((PQ)~R) 1 2 3 4 5 6 (formula)
P Q R ~R (PQ) ((PQ)~R)
1 T T2 T T3 T F4 T F5 F T6 F T7 F F8 F F
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Truth-table to evaluate ((PQ)~R) 1 2 3 4 5 6 (formula)
P Q R ~R (PQ) ((PQ)~R)
1 T T T2 T T F3 T F T4 T F F5 F T T6 F T F7 F F T8 F F F
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Truth-table to evaluate ((PQ)~R) 1 2 3 4 5 6 (formula)
P Q R ~R (PQ) ((PQ)~R)
1 T T T F2 T T F T3 T F T F4 T F F T5 F T T F6 F T F T7 F F T F8 F F F T
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Truth-table to evaluate ((PQ)~R) 1 2 3 4 5 6 (formula)
P Q R ~R (PQ) ((PQ)~R)
1 T T T F T2 T T F T T3 T F T F F4 T F F T F5 F T T F F6 F T F T F7 F F T F F8 F F F T F
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Truth-table to evaluate ((PQ)~R) 1 2 3 4 5 6 (formula)
P Q R ~R (PQ) ((PQ)~R)
1 T T T F T F2 T T F T T T3 T F T F F T4 T F F T F T5 F T T F F T6 F T F T F T7 F F T F F T8 F F F T F T
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Determining the logical status of (~PP)
1 2 3 (formula)P ~P (~PP)
12
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Determining the logical status of (~PP)
1 2 3 (formula)P ~P (~PP)
1 T2 F
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Determining the logical status of (~PP)
1 2 3 (formula)P ~P (~PP)
1 T F2 F T
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Determining the logical status of (~PP)
1 2 3 (formula)P ~P (~PP)
1 T F T2 F T F
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Determining the logical status of (~PP)
1 2 3 (formula)P ~P (~PP)
1 T F T2 F T F
Formula is CONTINGENT.
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Determining the logical status of (P~Q)
1 2 3 4P Q ~Q (P~Q)
1234
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Determining the logical status of (P~Q)
1 2 3 4P Q ~Q (P~Q)
1 T2 T3 F4 F
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Determining the logical status of (P~Q)
1 2 3 4P Q ~Q (P~Q)
1 T T2 T F3 F T4 F F
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Determining the logical status of (P~Q)
1 2 3 4P Q ~Q (P~Q)
1 T T F2 T F T3 F T F4 F F T
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Determining the logical status of (P~Q)
1 2 3 4P Q ~Q (P~Q)
1 T T F F2 T F T T3 F T F F4 F F T F
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Steps for determining logical status of a proposition.
• Create truth-table with right number of rows and columns.
• Compute truth-value of every formula on every row using truth-values of atomic statements for the row.
• Check for the logical status by checking the relevant rows.
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The logical status of a proposition.
Can use truth-tables to distinguish these different kinds of propositions.
Logical status Description Truth-Table
Tautology Always true Every row T
Contradiction Always false Every row F
Contingent Depends on circumstances
Some row T, some row F
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Determining the logical status of a group of propositions.
• Build truth-table to give side-by-side comparison of the truth values of the propositions.
• Doing this in a single truth-table is the best way to ensure an apples-to-apples comparison.
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Logical status of ~(PQ), (~P~Q)
1 2 3 4 5 6 7
P Q ~P ~Q (PQ) ~(PQ) (~P~Q)1
2
3
4
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Logical status of ~(PQ), (~P~Q)
1 2 3 4 5 6 7
P Q ~P ~Q (PQ) ~(PQ) (~P~Q)1 T2 T3 F4 F
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Logical status of ~(PQ), (~P~Q)
1 2 3 4 5 6 7
P Q ~P ~Q (PQ) ~(PQ) (~P~Q)1 T T2 T F3 F T4 F F
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Logical status of ~(PQ), (~P~Q)
1 2 3 4 5 6 7
P Q ~P ~Q (PQ) ~(PQ) (~P~Q)1 T T F2 T F F3 F T T4 F F T
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Logical status of ~(PQ), (~P~Q)
1 2 3 4 5 6 7
P Q ~P ~Q (PQ) ~(PQ) (~P~Q)1 T T F F2 T F F T3 F T T F4 F F T T
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Logical status of ~(PQ), (~P~Q)
1 2 3 4 5 6 7
P Q ~P ~Q (PQ) ~(PQ) (~P~Q)1 T T F F T2 T F F T F3 F T T F F4 F F T T F
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Logical status of ~(PQ), (~P~Q)
1 2 3 4 5 6 7
P Q ~P ~Q (PQ) ~(PQ) (~P~Q)1 T T F F T F2 T F F T F T3 F T T F F T4 F F T T F T
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Logical status of ~(PQ), (~P~Q)
1 2 3 4 5 6 7
P Q ~P ~Q (PQ) ~(PQ) (~P~Q)1 T T F F T F F2 T F F T F T T3 F T T F F T T4 F F T T F T T
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Logical status of ~(PQ), (~P~Q)
Both formulae have the same truth-values on every row (equivalent)
1 2 3 4 5 6 7
P Q ~P ~Q (PQ) ~(PQ) (~P~Q)1 T T F F T F F2 T F F T F T T3 F T T F F T T4 F F T T F T T
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Logical status of (PQ), (PQ)
1 2 3 4
P Q (PQ) (PQ)1
2
3
4
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Logical status of (PQ), (PQ)
1 2 3 4
P Q (PQ) (PQ)1 T2 T3 F4 F
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Logical status of (PQ), (PQ)
1 2 3 4
P Q (PQ) (PQ)1 T T2 T F3 F T4 F F
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Logical status of (PQ), (PQ)
1 2 3 4
P Q (PQ) (PQ)1 T T T2 T F F3 F T F4 F F F
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Logical status of (PQ), (PQ)
1 2 3 4
P Q (PQ) (PQ)1 T T T T2 T F F T3 F T F T4 F F F F
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Logical status of (PQ), (PQ)
A row (1) where both formulae are T (satisfiable, but not equivalent)
1 2 3 4
P Q (PQ) (PQ)1 T T T T2 T F F T3 F T F T4 F F F F
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Logical status of (P~Q), (PQ)
1 2 3 4
P Q ~Q (P~Q) (PQ)1
2
3
4
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Logical status of (P~Q), (PQ)
1 2 3 4
P Q ~Q (P~Q) (PQ)1 T2 T3 F4 F
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Logical status of (P~Q), (PQ)
1 2 3 4
P Q ~Q (P~Q) (PQ)1 T T2 T F3 F T4 F F
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Logical status of (P~Q), (PQ)
1 2 3 4
P Q ~Q (P~Q) (PQ)1 T T F2 T F T3 F T F4 F F T
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Logical status of (P~Q), (PQ)
1 2 3 4
P Q ~Q (P~Q) (PQ)1 T T F F2 T F T T3 F T F T4 F F T T
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Logical status of (P~Q), (PQ)
1 2 3 4
P Q ~Q (P~Q) (PQ)1 T T F F T2 T F T T F3 F T F T F4 F F T T F
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Logical status of (P~Q), (PQ)
NO row where both formulae are T (unsatisfiable)
1 2 3 4
P Q ~Q (P~Q) (PQ)1 T T F F T2 T F T T F3 F T F T F4 F F T T F
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Steps for determining logical status of group of propositions.
• Create truth-table with right number of rows and columns.
• Compute truth-value of every formula on every row using truth-values of atomic statements for the row.
• Check for the logical status by checking the relevant rows.
![Page 62: Truth table analysis](https://reader038.fdocuments.in/reader038/viewer/2022103004/55d505ddbb61eb4b338b4606/html5/thumbnails/62.jpg)
The logical status of a group of propositions.
Logical Status Description Truth-Table
Equivalent Mean the same thing, logically
The truth-tables are identical
Satisfiable/consistent
Possibly all true. There is some row where all propositions are true.
Unsatisfiable/inconsistent
Can’t all be true. No row where all propositions are true.
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Determining the validity of an argument via truth-tables.
• Remember that a valid argument is one where, if all the premises are true, the conclusion must be true.
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Determining the validity of an argument via truth-tables.
• Remember that a valid argument is one where, if all the premises are true, the conclusion must be true.
• Build truth table that includes formulae for premises and conclusion.
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Determining the validity of an argument via truth-tables.
• Remember that a valid argument is one where, if all the premises are true, the conclusion must be true.
• Build truth table that includes formulae for premises and conclusion.
• If there’s a row where the premises are true and the conclusion is false, argument is invalid.
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Determining the validity of an argument via truth-tables.
• Remember that a valid argument is one where, if all the premises are true, the conclusion must be true.
• Build truth table that includes formulae for premises and conclusion.
• If there’s a row where the premises are true and the conclusion is false, argument is invalid.
• Otherwise, argument is valid.
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Determining the validity of an argument via truth-tables.
1. If John wins the election, then Mary will run for Governor.
2. John did not win the election. (Conclusion) So, Mary will not run for Governor.
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Determining the validity of an argument via truth-tables.
1. If John wins the election, then Mary will run for Governor.
2. John did not win the election. (Conclusion) So, Mary will not run for Governor.
P = John wins the election.Q = Mary will run for Governor.
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Determining the validity of an argument via truth-tables.
1. If John wins the election, then Mary will run for Governor. (PQ)
2. John did not win the election. ~P(Conclusion) So, Mary will not run for Governor.
~Q
P = John wins the election.Q = Mary will run for Governor.
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Determining the validity of an argument via truth-tables.
Pr1 Pr2 C
P Q (PQ) ~P ~Q1
2
3
4
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Determining the validity of an argument via truth-tables.
Pr1 Pr2 C
P Q (PQ) ~P ~Q1 T2 T3 F4 F
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Determining the validity of an argument via truth-tables.
Pr1 Pr2 C
P Q (PQ) ~P ~Q1 T T2 T F3 F T4 F F
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Determining the validity of an argument via truth-tables.
Pr1 Pr2 C
P Q (PQ) ~P ~Q1 T T T2 T F F3 F T T4 F F T
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Determining the validity of an argument via truth-tables.
Pr1 Pr2 C
P Q (PQ) ~P ~Q1 T T T F2 T F F F3 F T T T4 F F T T
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Determining the validity of an argument via truth-tables.
Pr1 Pr2 C
P Q (PQ) ~P ~Q1 T T T F F2 T F F F T3 F T T T F4 F F T T T
![Page 76: Truth table analysis](https://reader038.fdocuments.in/reader038/viewer/2022103004/55d505ddbb61eb4b338b4606/html5/thumbnails/76.jpg)
Determining the validity of an argument via truth-tables.
Pr1 Pr2 C
P Q (PQ) ~P ~Q1 T T T F F2 T F F F T3 F T T T F4 F F T T T
ROW 3: Both premises true, conclusion false.Argument is invalid!
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Determining the validity of an argument via truth-tables.
1. If John wins the election, then Mary will run for Governor.
2. John won the election. (Conclusion) So, Mary will run for Governor.
P = John wins the election.Q = Mary will run for Governor.
![Page 78: Truth table analysis](https://reader038.fdocuments.in/reader038/viewer/2022103004/55d505ddbb61eb4b338b4606/html5/thumbnails/78.jpg)
Determining the validity of an argument via truth-tables.
1. If John wins the election, then Mary will run for Governor. (PQ)
2. John won the election. P(Conclusion) So, Mary will run for Governor.
Q
P = John wins the election.Q = Mary will run for Governor.
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Determining the validity of an argument via truth-tables.
Pr1 Pr2 C
P Q (PQ) P Q1
2
3
4
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Determining the validity of an argument via truth-tables.
Pr1 Pr2 C
P Q (PQ) P Q1 T2 T3 F4 F
![Page 81: Truth table analysis](https://reader038.fdocuments.in/reader038/viewer/2022103004/55d505ddbb61eb4b338b4606/html5/thumbnails/81.jpg)
Determining the validity of an argument via truth-tables.
Pr1 Pr2 C
P Q (PQ) P Q1 T T2 T F3 F T4 F F
![Page 82: Truth table analysis](https://reader038.fdocuments.in/reader038/viewer/2022103004/55d505ddbb61eb4b338b4606/html5/thumbnails/82.jpg)
Determining the validity of an argument via truth-tables.
Pr1 Pr2 C
P Q (PQ) P Q1 T T T2 T F F3 F T T4 F F T
![Page 83: Truth table analysis](https://reader038.fdocuments.in/reader038/viewer/2022103004/55d505ddbb61eb4b338b4606/html5/thumbnails/83.jpg)
Determining the validity of an argument via truth-tables.
Pr1 Pr2 C
P Q (PQ) P Q1 T T T T2 T F F T3 F T T F4 F F T F
![Page 84: Truth table analysis](https://reader038.fdocuments.in/reader038/viewer/2022103004/55d505ddbb61eb4b338b4606/html5/thumbnails/84.jpg)
Determining the validity of an argument via truth-tables.
Pr1 Pr2 C
P Q (PQ) P Q1 T T T T T2 T F F T F3 F T T F T4 F F T F F
![Page 85: Truth table analysis](https://reader038.fdocuments.in/reader038/viewer/2022103004/55d505ddbb61eb4b338b4606/html5/thumbnails/85.jpg)
Determining the validity of an argument via truth-tables.
Pr1 Pr2 C
P Q (PQ) P Q1 T T T T T2 T F F T F3 F T T F T4 F F T F F
ROW 1: Both premises true, conclusion true.Argument is valid!
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Important note for homework:
Some of the expressions you’ll be evaluating in truth-tables use alternate notation.
means ~ means means