

Philosophy 103: Introduction to Logic
How to Construct a Truth Table
Abstract: The general principles for the construction of truth tables are explained and illustrated.
 How to construct the guide columns:
 Write out the number of variables (corresponding to the number
of statements) in alphabetical order.
 The number of lines needed is 2^{n} where n is the number
of variables. (E. g., with three variables, 2^{3} = 8).
 Start in the righthand column and alternate T's
and F's until you run out of lines.
 Then move left to the next column and alternate pairs of
T's and F's until you run out of lines.
 Then continue to the next lefthand column and double the
numbers of T's and F's until completed.
 Example: construct a truth table for p
( q r )

Line
No.
↓

I, A
variables in alphabetical
order
↓ 




III, A
First
line
all T
→


p 
q 
r 
q r 
p
(q r) 
1 
T 
T 
T 
T 
T 
I, B
number of lines
= 2^{n}
→
III, D
half T's
half F's 
2 
T 
T 
F 
T 
T 
3 
T 
F 
T 
T 
T 
4 
T 
F 
F 
F 
T 
5 
F 
T 
T 
T 
T 
6 
F 
T 
F 
T 
T 
7 
F 
F 
T 
T 
T 
8 
F 
F 
F 
F 
F 

III, B
last
line
all F

↑
I, E
alternate
double of previous
T and F 
↑
I, D
alternate
pairs of
T and F

↑
I, C
alternate
T and F



 As check that the guide columns were done correctly:
 The first horizonal line will be all T's.
 The last horizonal line will be all F's.
 The leftmost column will be evenly divided; the first half
all T's and the second half all F's.

