The Language of Argument

1 2 6

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

Process of elimination is sometimes confused with a similar but crucially

different pattern of reasoning, which can be called affirming a disjunct. This

pattern includes both of these forms:

3. p ∨ q
   p
   ∴ ~q
   These forms of argument are invalid. This can be shown by the following
   single instance:
   She is either talking to herself or talking on a phone.
   She is talking to herself.
   ∴ She is not talking on a phone.
   This argument might seem valid if one assumes that she cannot talk on the
   phone while talking to herself. The premises, however, do not specify that
   she cannot do both at once. If she mumbles a few quick words to herself in
   the midst of talking on the phone, then the premises are both true and the
   conclusion is false.


4. p ∨ q
   q
   ∴ ~p

Explain why argument forms 1–2 are valid. Use common language that would

be understandable to someone who has not read this chapter.

Exercise X

Give other instances of argument forms 3–4 that are not valid. Explain why

these instances are invalid and why they show that the general argument form

is invalid.

Exercise XI

How Truth-Functional Connectives Work

We have now defined conjunction, disjunction, and negation. That, all by

itself, is sufficient to complete the branch of modern logic called proposi-

tional logic. The definitions themselves may seem peculiar. They do not

look like the definitions we find in a dictionary. But the form of these defi-

nitions is important, for it tells us something interesting about the charac-

ter of such words as “and,” “or,” and “not.” Two things are worth noting:

(1) These expressions are used to construct a new proposition from old ones;

(2) the newly constructed proposition is always a truth function of the origi-

nal propositions—that is, the truth value of the new proposition is always

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