We have seen that a compound statement is truth-functional: Its
truth or falsity is a function of the truth or falsity of its
component statements. For compound statements with just one
connective, the truth table for that connective tells us how to
make the computation. But we can construct truth tables for more
complex statements with more than one connective.
Consider the conditional statement A
(A
B). Because the main
connective is the conditional, it has the basic logical form pq, and its truth value is a function of p and q. But q in
this case is itself a compound statement, AB. So we need
to compute this element first.
In general, we work from the inside out, starting with the
connectives that apply directly to the components and doing the
main connective last.
| A |
B |
AB |
A(AB) |
| T |
T |
T |
T |
| F |
T |
F |
T |
| T |
F |
F |
F |
| F |
F |
F |
T |
If there are more components, we need more lines -- with three
components, for example, we need eight lines -- but the procedure
is the same. Let's construct the truth table for the
statement ~A(B v C).
| A |
B |
C |
~A |
(B v C) |
~A(B v C) |
| T |
T |
T |
F |
T |
F |
| T |
T |
F |
F |
T |
F |
| T |
F |
T |
F |
T |
F |
| T |
F |
F |
F |
F |
F |
| F |
T |
T |
T |
T |
T |
| F |
T |
F |
T |
T |
T |
| F |
F |
T |
T |
T |
T |
| F |
F |
F |
T |
F |
F |
To construct a truth table for a statement with more than one
connective:
1. Make a column for each component statement, with enough rows
for each possible combination of truth values among the
components.
2. Identify the connectives that apply directly to component
statements. On each row, determine the truth value of the
statement involving just that connective, and enter that truth
value in a column under the connective.
3. Repeat step #2 until you reach the main connective (the one
outside all parentheses). The truth values in its column are the
truth values of the statement as a whole.
Comprehension Questions