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

Both arguments are instances of the valid argument form “p; ∴ q ⊃ p.”

The other argument form is also paradoxical. It seems to say that a false

proposition implies any proposition whatsoever. The following is an in-

stance of this argument form:

Columbus was not president.

∴ If Columbus was president, then the moon is made of cheese.

Here it is hard to see what the falsehood that Columbus was president has to

do with the composition of the moon.

At this point, nonphilosophers become impatient, whereas philosophers

become worried. We started out with principles that seemed to be both ob-

vious and simple. Now, quite suddenly, we are being overwhelmed with a

whole series of peculiar results. What in the world has happened, and what

should be done about it? Philosophers remain divided in the answers they

give to these questions. The responses fall into two main categories: (1) Sim-

ply give up the idea that conditionals can be defined by truth-functional

techniques and search for a different and better analysis of conditionals that

avoids the difficulties involved in truth-functional analysis; or (2) take the

difficult line and argue that there is nothing wrong with calling the afore-

mentioned argument forms valid.

The first approach is highly technical and cannot be pursued in detail in

this book, but the general idea is this: Instead of identifying “If p, then q”

with “Not both p and not q,” identify it with “Not possibly both p and not q.”

This provides a stronger notion of a conditional and avoids some—though

not all—of the problems concerning conditionals. This theory is given a sys-

tematic development by offering a logical analysis of the notion of possibil-

ity. This branch of logic is called modal logic, and it has shown remarkable

development in recent decades.

The second line has been taken by Paul Grice, whose theories played a

prominent part in Chapter 2. He acknowledges—as anyone must—that the

two argument forms above are decidedly odd. He denies, however, that this

oddness has anything to do with validity. Validity concerns one thing and

one thing only: a relationship between premises and conclusion. An argu-

ment is valid if the premises cannot be true without the conclusion being

true as well. The above arguments are valid by this definition of “validity.”

Of course, arguments can be defective in all sorts of other ways. Look

at the first argument form: (1) p; ∴ q ⊃ p. Because “q” can be replaced by

any proposition (true or false), the rule of Relevance will often be violated.

It is worth pointing out violations of the rule of Relevance, but, according

to Grice, this issue has nothing to do with validity. Beyond this, arguments

having this form can also involve violations of the rule of Quantity. A con-

ditional will be true whenever the consequent is true. Given this, it does not

matter to the truth of the whole conditional whether the antecedent is true

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