Now that we understand how to put statements into symbolic notation, it's time to look at arguments. In this section, we will learn how to use truth tables to determine whether an argument is valid.

A valid argument is one in which the truth of the premises would guarantee the truth of the conclusion. The premises need not actually be true. What matters for validity is the relation between the premises and the conclusion. In a valid argument, the relation is such that if the premises are true, the conclusion must be true as well.

We have seen that the truth or falsity of a compound statement involving one of the connectives is determined by the truth or falsity of its components, according to a truth table. This means we can also use truth tables to assess the validity of arguments involving compound statements.

Consider the inference form modus tollens:

p q ~ q ~ p

To show that it is valid, we construct a truth table with columns for the components, columns for the premises, and a column for the conclusion.

Components		Premises		Conclusion
p	q	p q	~ q	~ p
T	T	T	F	F
F	T	T	F	T
T	F	F	T	F
F	F	T	T	T

The truth values assigned to the components p and q determine the truth value of each premise and of the conclusion; all three propositions in the argument have truth values dependent on the same component propositions. The test for validity is whether the conclusion can be false if the premises are both true. So we look first at the column for the conclusion: ~ p is false in two cases, the first and third lines. However, in the first line, the premise ~ q is false, and in the third line, the premise p q is false. So there is no case in which the conclusion is false and both premises are true.

Alternatively, we could start with the premises. There is only one case--the fourth line--in which they are both true, and in that case, the conclusion is true as well.

Now consider an invalid argument that affirms the consequent:

C D D C

Once again we construct a truth table:

Components		Premises		Conclusion
C	D	C D	D	C
T	T	T	T	T
F	T	T	T	F
T	F	F	F	T
F	F	T	F	F

As before, we start with the components on the left, then the premises in order, and finally the conclusion on the right. Since the second premise and the conclusion are the components themselves, these columns simply repeat the ones on the left. Once you get used to this method, you can simplify things by skipping the repetition. In any case, we can see why the argument is invalid. In the second line, both premises are true and the conclusion is false.