PREMISE PREMISE CONCLUSION
p q r (p q) (p r) ~(p r) q
T T T T T F T
T T F T T F T
T F T T T F F
T F F T T F F
F T T T T F T
F T F T F T T OK
F F T F T F F
F F F F F T F
Notice that there is only one combination of truth values for which both premises are true, and in that case the conclusion is true as well. So the origi- nal argument is valid because it is an instance of a valid argument form—that is, an argument form with no substitution instances for which true premises are combined with a false conclusion.
This last truth table may need some explaining. First, why do we get eight rows in this truth table where before we got only four? The answer to this is that we need to test the argument form for every possible combination of truth values for the component propositions. With two variables, there are four possible combinations: (TT), (TF), (FT), and (FF). With three variables, there are eight possible combinations: (TTT), (TTF), (TFT), (TFF), (FTT), (FTF), (FFT), and (FFF). The general rule is this: If an argument form has n vari- ables, the truth table used in its analysis must have 2n rows. For four vari- ables there will be sixteen rows; for five variables, thirty-two rows; for six variables, sixty-four rows; and so on. You can be sure that you capture all possible combinations of truth values by using the following pattern in con- structing the columns of your truth table under each individual variable:
First column Second column Third column . . .
First half Ts, First quarter Ts, First eighth Ts,
second half Fs. second quarter Fs, second eighth Fs,
and so on. and so on.
A glance at the earlier examples in this chapter will show that we have been using this pattern, and it is the standard way of listing the possibilities.
p q ~(p r)
q
The expression on the right gives the argument form of the argument pre- sented on the left. To test the argument for validity, we ask whether the argu- ment form is valid. The procedure is cumbersome, but perfectly mechanical:
