UNDERSTANDING TRUTH TABLES
Since any
truth-functional proposition changes
its value as the variables change, we should get some idea of what
happens
when we change these values systematically. Once again we will use a
red
background for something true and a blue background for something
false.
This time we will use example statements (or propositions) that would
be
represented by letters such as "P" or "Q."
First we
have a conjunction, true only when
both parts are true.
| logic
is easy |
and |
everyone
studies |
| T |
T |
T |
| T |
F |
F |
| F |
F |
T |
| F |
F |
F |
| P |
& |
Q |
| T |
T |
T |
| T |
F |
F |
| F |
F |
T |
| F |
F |
F |
Next we
will see what happens with (inclusive)
disjunction, when both parts must be false in order for the proposition
itself to be false.
| (either)
logic is easy |
or |
everyone
studies |
| T |
T |
T |
| T |
T |
F |
| F |
T |
T |
| F |
F |
F |
| P |
v |
Q |
| T |
T |
T |
| T |
T |
F |
| F |
T |
T |
| F |
F |
F |
We then
have the relationship of implication,
when a proposition expressing a condition is false only at the time it
has a true antecedent and a false consequent.
| logic
is easy |
only
if |
everyone
studies |
| T |
T |
T |
| T |
F |
F |
| F |
T |
T |
| F |
T |
F |
| P |
-> |
Q |
| T |
T |
T |
| T |
F |
F |
| F |
T |
T |
| F |
T |
F |
Finally
we have the relationship of equivalence
or mutual implication (the opposite of mutual exclusion), when the only
way a proposition can be false is for it to have different truth values
for its parts.
| logic
is easy |
if
and only if |
everyone
studies |
| T |
T |
T |
| T |
F |
F |
| F |
F |
T |
| F |
T |
F |
| P |
<-> |
Q |
| T |
T |
T |
| T |
F |
F |
| F |
F |
T |
| F |
T |
F |
Next, we go
on to see how to use a truth
table to decide the validity of an argument form. Click here to
go ahead, or click to go back to the
explanation
of truth-functional propositions.