USING TRUTH TABLES TO TEST FOR
VALIDITY
The four
connectives (or operators) we have
used so far are all binary, meaning they link two distinct
elements. There is also the unary operator for negation (NOT).
Its effect is to reverse the original value of a proposition. For
simplicity, we will not isolate this operator in the following charts
but show its effect in how we color the areas.
The
principal use of truth tables in
symbolic logic is to test for deductive validity. What this
means is that we are trying to determine whether there even is the
possibility that a combination of true premises could imply a false
conclusion. See the difference in the two examples below. The first
represents a pattern that is valid, and the main connective is always
"true." The second represents an invalid pattern, which once has a
"false" main connective. (Think of the main connective--the one for
implication, in this case--as carrying the truth value of the entire
complex proposition.)
| (That) logic is easy |
only if |
everyone studies |
but |
not everyone studies |
implies |
logic is not easy. |
| T (logic is easy) |
T |
T (everyone studies) |
F |
F (everyone studies) |
T |
F (logic is
easy) |
| T |
F |
F |
F |
T |
T |
F |
| F |
T |
T |
F |
F |
T |
T |
| F |
T |
F |
T
|
T |
T |
T |
| ((P |
-> |
Q) |
& |
~Q) |
-> |
~P |
| T |
T |
T |
F |
F |
T |
F |
| T |
F |
F |
F |
T |
T |
F |
| F |
T |
T |
F |
F |
T |
T |
| F |
T |
F |
T
|
T |
T |
T |
| (That) logic is easy |
only if |
everyone studies |
and |
everyone studies |
implies |
logic is easy. |
| T (logic is easy) |
T |
T (everyone studies) |
T |
T (everyone studies) |
T |
T (logic is
easy) |
| T |
F |
F |
F |
F |
T |
T |
| F |
T |
T |
T |
T |
F |
F |
| F |
T |
F |
F |
F |
T |
F |
| ((P |
-> |
Q) |
& |
Q) |
-> |
P |
| T |
T |
T |
T |
T |
T |
T |
| T |
F |
F |
F |
F |
T |
T |
| F |
T |
T |
T |
T |
F |
F |
| F |
T |
F |
F |
F |
T |
F |
Either go
forward to how to complete a truth table or
go back and review the use of connectives.