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Logic: Truth Tables
Constructing a Truth Table
Truth Table
A truth table for a compound statement is a list of the truth or falsity of the statement for every possible combination of truth and falsity of its components.
In other words, a truth table helps to show whether a statement is true or false.
Rows
To find the number of rows used in a truth table, take the number 2 raised to the power of the number of variables.
For example, if there was a p statement and a q statement, there would be 2 variables, 2^2 is 4.
If there were three statements, it would be 2^3, or 8 rows.
Columns
The columns under the connectives /\, and \/, stand for the conjunction, and disjunction of the expression on the two sides of that connective.
Two statement table
p q
THalf the rows should be true
TThe rows should
alternate T, F
T FThe result should
be
FHalf the rows
should be falseT
one of every possibility
F F TT, TF, FT, FF
The three statement tablep q r
T Half of the rows T Alternate TT T Alternate
T should be true, T and FF F T and F
T the other half F T so that there is
T should be false. F F one of every
F T T possibility
F T F TTT, TTF, TFT
F F T ect.
F F F
Negation Truth Table
p ~ p The opposite of p is ~p
T F “Not true” is “false”
F T “Not false” is “true”
Conjunction Truth Table
p q p /\ q p and q
T T TTrue only if
both are true.
T F F
F T F
F F F
Disjunction Truth Table
p q p \/ q p or q
T T TTrue if either
on is true
T F T
F T T
F F FFalse only if both are false
Lets fill out a table
p q p \/(or)
(~p(not)
/\(and)
q)
T T
T F
F T
F F
Negate the p column
p q p \/(or)
(~p(not)
/\(and)
q)
T T F
T F F
F T T
F F T
Copy the q column
p q p \/(or)
(~p(not)
/\(and)
q)
T T F T
T F F F
F T T T
F F T F
Fill the /\ column
p q p \/(or)
(~p(not)
/\(and)
q)
T T F F T
T F F F F
F T T T T
F F T F F
Copy the p column
p q p \/(or)
(~p(not)
/\(and)
T T T F F
T F T F F
F T F T T
F F F T F
Fill in the \/ column using the p and the /\ columns
p q P \/(or)
P and (~p/\p)
(~p(not)
/\(and)
(~p) and (p)
p
T T T T F F T
T F T T F F T
F T F T T T F
F F F F T F F
Use the final column to determine what type of statements it is
\/(or)
P and (~p/\p)
Tautology
Always True
Contradiction
Always False
ContingencySometimes true, sometimes false
T x x
T x x
T x x
F x x
Contingency
Some were true, while one was false. That makes this statement a contingency.
Real life example In case that was not entirely clear, let’s take a look
at an everyday example. Circuits. There are two different kinds of circuits,
a series circuit and a parallel circuit. When the switch is closed the light will be on.
However, with a series circuit, both switches have to be closed and with a parallel circuit only one switch has to be closed for the light to go on.
Series Circuits
Switch p Switch q Light
Closed Closed On Only on if both are closed
Closed Open Off
Open Closed Off
Open Open Off
Parallel Circuits
Switch p Switch q Light
Closed Closed On
Closed Open On
Open Closed On
Open Open Off Only off when both are open
Conclusion
That concludes the Logic: Truth Tables lesson. For more information, consult Finite Mathematics
by Berresford and Rockett. Or learn logic online:
http://www.earlham.edu/~peters/courses/log/terms2.htm
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