Truth-Table Example
1
Truth Tables – A Step-By-Step Example
In what follows, the latest additions to the table are given in boldface.
Suppose we are asked to do the truth table for the following statement form:
(1) Q v (P Æ (~Q v ~P))
What follows is a breakdown of the process of building the table for (1).
Step 1: Create the blank table.
There are two atomic statements, P and Q, so the table will have four rows and three columns,
like so (see end of document for more information):
P
T
T
F
F
Q
T
F
T
F
Q v (P Æ (~Q v ~P))
Step 2: Determine the main connective.
The main connective of (1) is the first occurrence of disjunction (“v”). So the column under this
instance of disjunction will be our ‘final answer’ – our goal is to fill in this column.
Now, in order to figure out what belongs in that main column (under the main connective), we
need to figure out the values for the two disjuncts.
Step 3: Fill in the table.
Figuring out the value for the left disjunct (Q) is easy, because we already have it listed to the
left. So we can simply copy the values over:
P
T
T
F
F
Q
T
F
T
F
Q v (P Æ (~Q v ~P))
T
F
T
F
Turning now to the right disjunct, we see that it is a compound statement, “P Æ (~Q v ~P)”.
Since it is a conditional, we need to figure out the values for its antecedent and consequent. The
antecedent is easy – it’s just P, and we have the values for P to the left. The consequent is a
disjunction, we need to figure out the values for its two disjuncts, “~Q” and “~P”. Both of these
are negations, so they simply switch True to False and vice versa.
Truth-Table Example
So, first, copy the values of Q and P from the left hand columns:
P
T
T
F
F
Q
T
F
T
F
Q v (P --> (~Q v ~P))
T
T
T
T
F
T
F
T
T
F
T
F
F
F
F
F
Next, figure out the values of the two disjuncts in the consequent of the conditional:
P
T
T
F
F
Q
T
F
T
F
Q v (P --> (~Q v ~P))
T
T
FT
FT
F
T
TF
FT
T
F
FT
TF
F
F
TF
TF
Now we have the information we need to determine the values for the disjunction in the
consequent of the conditional. Consulting the characteristic truth table for disjunction, we get:
P
T
T
F
F
Q
T
F
T
F
Q v (P --> (~Q v ~P))
T
T
FT F FT
F
T
TF T FT
T
F
FT T TF
F
F
TF T TF
So far, so good. We now have the values for both the antecedent and the consequent of the
conditional, so, consulting the characteristic truth table for conditionals, we can fill in the values
for the conditional:
P
T
T
F
F
Q
T
F
T
F
Q v (P --> (~Q v ~P))
T
T F
FT F FT
F
T T
TF T FT
T
F T
FT T TF
F
F T
TF T TF
Finally, we have all the information we need to figure out the values for the main column:
P
T
T
F
F
Q
T
F
T
F
Q
T
F
T
F
v (P --> (~Q v ~P))
T T F
FT F FT
T T T
TF T FT
T F T
FT T TF
T F T
TF T TF
So, it turns out that any statement with this form is always true.
2
Truth-Table Example
ADDENDUM
Since each atomic statement has two possible values (True or False), a truth table will have 2n
rows, where n is the number of atomic statements. So, if there are two atomic statements, the
table has four rows; three atomic statements requires eight rows; four requires 16 rows; and so
forth.
So, for example, here is a blank eight-row table:
T
T
T
T
F
F
F
F
T
T
F
F
T
T
F
F
T
F
T
F
T
F
T
F
The three left-most columns list every possible permutation of Ts and Fs.
3