The Language of Argument

1 2 0

C H A P T E R 6 ■ P r o p o s i t i o n a l L o g i c

The first two columns give all the combinations for the truth values of the

propositions that we might substitute for “p” and “q.” The third column

gives the truth value of the premise for each of these combinations. (This

column is the same as the definition for “&” given above.) Finally, the fourth

column gives the truth value for the conclusion for each combination. (Here,

of course, this merely involves repeating the first column. Later on, things

will become more complicated and interesting.) If we look at this truth table,

we see that no matter how we make substitutions for the variables, we never

have a case in which the premise is true and the conclusion is false. In the

first line, the premise is true and the conclusion is also true. In the remaining

three lines, the premise is not true, so the possibility of the premise being

true and the conclusion false does not arise.

Here it is important to remember that a valid argument can have false

premises, for one proposition can follow from another proposition that is

false. Of course, an argument that is sound cannot have a false premise,

because a sound argument is defined as a valid argument with true premises.

But our subject here is validity, not soundness.

Let’s summarize this discussion. In the case we have examined, validity

depends on the form of an argument and not on its particular content. A first

principle, then, is this:

An argument is valid if it is an instance of a valid argument form.

Hence, the argument “Harry is short and John is tall; therefore, Harry is

short” is valid because it is an instance of the valid argument form “p & q;

∴ p.”

Next we must ask what makes an argument form valid. The answer to

this is given in this principle:

An argument form is valid if and only if it has no substitution instances

in which the premises are true and the conclusion is false.

We have just seen that the argument form “p & q; ∴ p” passes this test. The

truth table analysis showed that. Incidentally, we can use the same truth

table to show that the following argument is valid:

John is tall. p

Harry is short. q

∴ John is tall and Harry is short. ∴ p & q

The argument on the left is a substitution instance of the argument form on

the right. A glance at the truth table will show that there can be no cases for

which all the premises could be true and the conclusion false. This pretty

well covers the logical properties of conjunction.

Notice that we have not said that every argument that is valid is so in

virtue of its form. There may be arguments in which the conclusion follows

from the premises but we cannot show how the argument’s validity is a

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