ADDITIONAL
MATERIAL ON TRUTH-TABLE TESTING
When an argument form is valid
we know that we could not have a
situation in which the conclusion is false unless one or more premises
would be false as well. A direct test for this involves
setting up parallel truth tables for each premise and for the
conclusion. To make the task easier we also set up separate
columns for each variable. The number of lines depends on the the
number of variables (2n), so that with three variables we
would have eight lines. Below is an example of an invalid form,
one in which there is what we call a "bad line"--a situation in which
all the premises are true but the conclusion is false.
If logic is
easy and fun then it is interesting. Logic is not
interesting. Therefore, logic is not easy.
(E & F) -> I, I- ~E
E F I
|| (E & F) -> I | ~I | ~E
T T
T
T T T F F
T T
F
T F F T F
T F
T
F T T F F
T F
F
F T F T
F - a "bad line" (true premises but a false conclusion)
F T
T
F T T F T
F T
F
F T F T T
F F
T
F T T F T
F F
F
F T F T T
Because a long truth table is
tedious to complete, you are strongly recommended to master the
indirect or reverse method, which involves setting up what would count
as a bad line and then seeing if it is possible to fill in the
variables consistently. For instance, for the argument form
above we would have the following:
If logic is easy
and fun then it is interesting. Logic is not interesting.
Therefore, logic is not easy.
E F I
|| (E & F) -> I | ~I | ~E
T F F
F T F T
F invalid
here we can force the bad line (and often
there can be more than one way of doing this)
If we break down what we are doing
into a series of steps, we can see our progress this way:
E
F I
|| (E & F) -> I | ~I | ~E
? ? ?
T T
F step 1
T ? F T F
T F TF FT
step 2
T F F T F F
T F TF FT
step 3
With a valid argument form we discover that we cannot do this; at least
one variable would have to be both true and false at the same time, and
we can indicate this with a "X" below the variable in question.
For instance:
E F I ||
(E v F) -> I | ~I | ~E
T x F
F T F
T F valid
as long as I is false, the only way the
first premise could be true would be to have E v F false as well, but with E
true we cannot fill in a value for F
to make this happen