Completing truth tables of more than
four lines by hand is tedious and prone to error. Fortunately,
there is a very fast way of obtaining the same results. This is
by use of what we will call the reverse
method. It relies on the concept that for an invalid
argument form there will be at least one bad line--one line on which we
find the premises to be true while the conclusion is false.
By using the reverse method we attempt to see if we can in fact make
such a line occur.
Let's look
again a the last two argument forms that you tested.
1. (P & Q) ->
R, ~R |- ~P
P
Q
R
(P
& Q) -> R
~R
~P
1
1
1
1
0
0
1
1
0
0
1
0
1
0
1
1
0
0
1
0
0
1
1
0
0
1
1
1
0
1
0
1
0
1
1
1
0
0
1
1
0
1
0
0
0
1
1
1
Invalid
(you have a bad line when P is true but both Q and R are false)
To use the reverse method we will
set up our pattern as we have before with the letters in use at the
beginning and then the expressions making up the premises and the
conclusion. Note that to minimize confusion we will use a double
bar to separate the letters and single bars to separate the expressions
in the form itself.
P
Q R || (P & Q) -> R | ~R | ~P
Next
we will set the premises to be true and the conclusion false.
P Q R
|| (P & Q) -> R | ~R | ~P
1 1 0
Next, we will work backwards to see
how we can account for the three letters to the left in keeping with
what we first know. We can immediately see the values for P and R.
P
Q R || (P & Q) -> R | ~R | ~P 1 0
1 1 0
The
task now is to see whether we can also account for the value of
Q. What we find is that
having Q false does fit,
and so we have the bad line that lets us say the form is invalid. P
Q R || (P & Q) -> R | ~R | ~P 1 0 0
1 1 0
invalid
2. (P v Q) -> R, ~R |- ~P
P
Q
R
(P
v Q) -> R
~R
~P
1
1
1
1
0
0
1
1
0
0
1
0
1
0
1
1
0
0
1
0
0
0
1
0
0
1
1
1
0
1
0
1
0
0
1
1
0
0
1
1
0
1
0
0
0
1
1
1
valid
(there are no bad lines)
Here again we will repeat the first
steps we followed above.
P Q R || (P v Q) -> R | ~R | ~P
1
0
1 1 0
Now, however, we hit a roadblock
that prevents us from moving any further. We cannot have Q either true or false since
with P already true the
entire expression P v Q
remains true. The "x" indicates the roadbloack and justifies our
finding that the argument form indeed is valid.
P Q R || (P v Q) -> R | ~R | ~P
1 x
0
1 1 0
valid
EXERCISES
(ON YOUR OWN) Use the reverse method to test the
following argument forms for
validity. 1. P -> (Q v R), S -> P, S
& ~Q |- R 2. P -> (Q v R), P ->
S, ~S |- ~(Q v R) 3. (P -> Q) v (R -> S), P v
R |- Q v S 4. (P & Q) -> (R v S), P, ~R
|- ~Q v S
Next
we will set the premises to be true and the conclusion false.