The Language of Argument

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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

Truth Tables for Conditionals

For conjunction, disjunction, and negation, the truth table method provides

an approach that is at once plausible and effective. A propositional condi-

tional is also compounded from two simpler propositions, and this suggests

that we might be able to offer a truth table definition for these conditionals

as well. What should the truth table look like? When we try to answer this

question, we get stuck almost at once, for it is unclear how we should fill in

the table in three out of four cases.

p q If p, then q

T T?

T F F

F T?

F F?

It seems obvious that a conditional cannot be true if the antecedent is true

and the consequent is false. We record this by putting “F” in the second row.

But suppose “p” and “q” are replaced by two arbitrary true propositions—

say, “Two plus two equals four” and “Chile is in South America.” Consider

what we shall say about the conditional:

If two plus two equals four, then Chile is in South America.

This is a very strange statement, because the arithmetical remark in the ante-

cedent does not seem to have anything to do with the geographical remark

in the consequent. So this conditional is odd—indeed, extremely odd—but

is it true or false? At this point, a reasonable response is bafflement.

Consider the following argument, which is intended to solve all these

problems by providing reasons for assigning truth values in each row of the

truth table. First, it seems obvious that, if “If p, then q” is true, then it is not the

case that both “p” is true and “q” is false. That in turn means that “~(p & ~q)”

must be true. The following, then, seems to be a valid argument form:

If p, then q.

∴~(p & ~q)

Second, we can also reason in the opposite direction. Suppose we know that

“~(p & ~q)” is true. For this to be true, “p & ~q” must be false. We know this

from the truth table definition of negation. Next let us suppose that “p” is

true. Then “~q” must be false. We know this from the truth table definition

of conjunction. Finally, if “~q” is false, then “q” itself must be true. This line

of reasoning is supposed to show that the following argument form is valid:

~(p & ~q)

∴If p, then q.

The first step in the argument was intended to show that we can validly derive

“~(p & ~q)” from “If p, then q.” The second step was intended to show that the

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