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Truth-Table Analysis
Phil 57 section 3San Jose State University
Fall 2010
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.
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.
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.
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
The logical status of a proposition.
Consider P~P
P ~P P~P
The logical status of a proposition.
Consider P~P
P ~P P~P
T
F
The logical status of a proposition.
Consider P~P
P ~P P~P
T F
F T
The logical status of a proposition.
Consider P~P
P ~P P~P
T F F
F T F
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
The logical status of a proposition.
Consider P~P
P ~P P~P
The logical status of a proposition.
Consider P~P
P ~P P~P
T
F
The logical status of a proposition.
Consider P~P
P ~P P~P
T F
F T
The logical status of a proposition.
Consider P~P
P ~P P~P
T F T
F T T
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
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?
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.
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.
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
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
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
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
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
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
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
Determining the logical status of (~PP)
1 2 3 (formula)P ~P (~PP)
12
Determining the logical status of (~PP)
1 2 3 (formula)P ~P (~PP)
1 T2 F
Determining the logical status of (~PP)
1 2 3 (formula)P ~P (~PP)
1 T F2 F T
Determining the logical status of (~PP)
1 2 3 (formula)P ~P (~PP)
1 T F T2 F T F
Determining the logical status of (~PP)
1 2 3 (formula)P ~P (~PP)
1 T F T2 F T F
Formula is CONTINGENT.
Determining the logical status of (P~Q)
1 2 3 4P Q ~Q (P~Q)
1234
Determining the logical status of (P~Q)
1 2 3 4P Q ~Q (P~Q)
1 T2 T3 F4 F
Determining the logical status of (P~Q)
1 2 3 4P Q ~Q (P~Q)
1 T T2 T F3 F T4 F F
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
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
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.
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
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.
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
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
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
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
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
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
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
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
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
Logical status of (PQ), (PQ)
1 2 3 4
P Q (PQ) (PQ)1
2
3
4
Logical status of (PQ), (PQ)
1 2 3 4
P Q (PQ) (PQ)1 T2 T3 F4 F
Logical status of (PQ), (PQ)
1 2 3 4
P Q (PQ) (PQ)1 T T2 T F3 F T4 F F
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
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
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
Logical status of (P~Q), (PQ)
1 2 3 4
P Q ~Q (P~Q) (PQ)1
2
3
4
Logical status of (P~Q), (PQ)
1 2 3 4
P Q ~Q (P~Q) (PQ)1 T2 T3 F4 F
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
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
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
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
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
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.
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.
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.
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.
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.
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.
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.
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.
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.
Determining the validity of an argument via truth-tables.
Pr1 Pr2 C
P Q (PQ) ~P ~Q1
2
3
4
Determining the validity of an argument via truth-tables.
Pr1 Pr2 C
P Q (PQ) ~P ~Q1 T2 T3 F4 F
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
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
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
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
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!
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.
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.
Determining the validity of an argument via truth-tables.
Pr1 Pr2 C
P Q (PQ) P Q1
2
3
4
Determining the validity of an argument via truth-tables.
Pr1 Pr2 C
P Q (PQ) P Q1 T2 T3 F4 F
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
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
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
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
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!
Important note for homework:
Some of the expressions you’ll be evaluating in truth-tables use alternate notation.
means ~ means means
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