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.