11.  TESTING FOR VALIDITY USING THE REVERSE METHOD

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

red flagNext 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

Click on for the answer key.