1. Constructing a Truth Table
Logic: Truth Tables
Constructing a Truth Table

2. 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.

3. 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.

4. Columns
The columns under the connectives /\, and \/, stand for the conjunction, and disjunction of the expression on the two sides of that connective.

5. Two statement table p q T Half the rows should be true
The rows should alternate T, F F
The result should be
Half the rows should be false
one of every possibility
TT, TF, FT, FF

6. The three statement tablep q r T
Half of the rows
Alternate TT
Alternate
should be true,
and FF F
T and F
the other half
so that there is
should be false.
one of every
possibility
TTT, TTF, TFT
ect.

7. Negation Truth Table p ~ p The opposite of p is ~p T F
“Not true” is “false”
“Not false” is “true”

8. Conjunction Truth Tablep q
p /\ q
p and q T
True only if both are true. F

9. Disjunction Truth Tablep q
p \/ q
p or q T
True if either on is true F
False only if both are false

10. Lets fill out a table p q \/
(or)
(~p
(not) /\
(and) q) T F

11. Negate the p column p q \/
(or)
(~p
(not) /\
(and) q) T F

12. Copy the q column p q \/
(or)
(~p
(not) /\
(and) q) T F

13. Fill the /\ column p q \/
(or)
(~p
(not) /\
(and) q) T F

14. Copy the p column p q \/
(or)
(~p
(not) /\
(and) T F

15. Fill in the \/ column using the p and the /\ columnsq P \/
(or)
P and (~p/\p)
(~p
(not) /\
(and)
(~p) and (p) T F

16. Use the final column to determine what type of statements it is\/
(or)
P and (~p/\p)
Tautology
Always True
Contradiction
Always False
Contingency
Sometimes true, sometimes false T x F

17. Contingency Some were true, while one was false.
That makes this statement a contingency.

18. 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.

19. Only on if both are closed
Series Circuits
Switch p
Switch q
Light
Closed On
Only on if both are closed
Open
Off

20. Only off when both are open
Parallel Circuits
Switch p
Switch q
Light
Closed On
Open
Off
Only off when both are open

21. Conclusion That concludes the Logic: Truth Tables lesson.
For more information, consult Finite Mathematics by Berresford and Rockett.
Or learn logic online: