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

Of course, as soon as an argument becomes at all complex, these truth tables

become very large indeed. But there is no need to worry about this, because

we will not consider arguments with many variables. Those who do so turn

to a computer for help.

The style of the truth table above is also significant. The premises are

plainly labeled, and so is the conclusion. A line is drawn under every row

in which the premises are all true. (In this case, there is only one such row—

row 6.) If the conclusion on this line is also true, it is marked “OK.” If every

line in which the premises are all true is OK, then the argument form is

valid. Marking all this may seem rather childish, but it is worth doing. First,

it helps guard against mistakes. More importantly, it draws one’s atten-

tion to the purpose of the procedure being used. Cranking out truth tables

without understanding what they are about—or even why they might be

helpful—does not enlighten the mind or elevate the spirit.

For the sake of contrast, we can next consider an invalid argument:

(1) Valerie is either a doctor or a lawyer.

(2) Valerie is not both a lawyer and a stockbroker.

∴(3) Therefore, Valerie is a doctor.

Using the same abbreviations as earlier, this becomes:

D ∨ L p ∨ q

~(L & S)^ ~(q & r)

∴ D ∴ p

The truth table for this argument form looks like this:

Premise Premise Conclusion

p q r (p ∨ q) (q & r) ~(q & r) p

T T T T T F T

T T F T F T T OK

T F T T F T T OK

T F F T F T T OK

F T T T T F F

F T F T F T F Invalid

F F T F F T F

F F F F F T F

This time, we find four rows in which all the premises are true. In three cases

the conclusion is true as well, but in one of these cases (row 6), the conclu-

sion is false. This line is marked “Invalid.” Notice that every line in which all

of the premises are true is marked either as “OK” or as “Invalid.” If even one

row is marked “Invalid,” then the argument form as a whole is invalid. The

argument form under consideration is thus invalid, because it is possible for

it to have a substitution instance in which all the premises are true and the

conclusion is false.

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