The Language of Argument

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C o n d i t i o n a l s

We can next look at sentences of the form “p if and only if q”—so-called

biconditionals. If I say that I will clean the barn if and only if Hazel will help

me, then I am saying that I will clean it if she helps and I will not clean it if

she does not. Translated, this becomes:

(H ⊃ B) & (~H ⊃ ~B)

This is equivalent to:

(H ⊃ B) & (B ⊃ H)

We thus have an implication going both ways—the characteristic form of a

biconditional. In fact, constructions containing the expression “if and only

if” do not often appear in everyday speech. They appear almost exclusively

in technical or legal writing. In ordinary conversation, we capture the force

of a biconditional by saying something like this:

I will clean the barn, but only if Hazel helps me.

The decision whether to translate a remark of everyday conversation into a

conditional or a biconditional is often subtle and difficult. We have already

noticed that the use of sentences of the form “p only if q” will often conver-

sationally imply a commitment to the biconditional “p if and only if q.” In

the same way, the use of the conditional “p if q” will often carry this same

implication. If I plan to clean the barn whether Hazel helps me or not, it will

certainly be misleading—again, a violation of the rule of Quantity—to say

that I will clean the barn if Hazel helps me.

We can close this discussion by considering one further, rather difficult

case. What is the force of saying “p unless q”? Is this a biconditional, or just

a conditional? If it is just a conditional, which way does the implication go?

There is a strong temptation to treat this as a biconditional, but the following

example shows this to be wrong:

To appreciate the complexities of the little word “only,” it is useful to notice

that it fits at every point in the sentence “I hit him in the eye”:

Only I hit him in the eye.

I only hit him in the eye.

I hit only him in the eye.

I hit him only in the eye.

I hit him in only the eye.

I hit him in the only eye.

I hit him in the eye only.

Explain what each of these sentences means.

Exercise XXVI

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