The Language of Argument

1 4 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

the first sentence is symbolized as follows:

H ⊃ B

Notice that in the prose version of item 1, the antecedent and consequent ap-

pear in reverse order; “q if p” means the same thing as “If p, then q.”

How shall we translate the second sentence? Here we should move

slowly and first notice what seems incontestable: If Hazel does not help

me, then I will not clean the barn. This is translated in the following

way:

~H ⊃ ~B

And that is equivalent to:

B ⊃ H

If this equivalence is not obvious, it can quickly be established using a truth

table.

A more difficult question arises when we ask whether an implication runs

the other way. When I say that I will clean the barn only if Hazel will help

me, am I committing myself to cleaning the barn if she does help me? There

is a strong temptation to answer the question “yes” and then give a fuller

translation of item 2 in the following way:

(B ⊃ H) & (H ⊃ B)

Logicians call such two-way implications biconditionals, and we will

discuss them in a moment. But adding this second conjunct is almost

surely a mistake, for we can think of parallel cases where we would

not be tempted to include it. A government regulation might read as

follows:

A student may receive a New York State Scholarship only if the student

attends a New York State school.

From this it does not follow that anyone who attends a New York State

school may receive a New York State Scholarship. There may be other

requirements as well—for example, being a New York State resident.

Why were we tempted to use a biconditional in translating sentences

containing the connective “only if”? Why, that is, are we tempted to think

that the statement “I’ll clean the barn only if Hazel will help me” implies “If

Hazel helps me, then I will clean the barn”? The answer turns on the notion

of conversational implication first discussed in Chapter 2. If I am not going

to clean the barn whether Hazel helps me or not, then it will be misleading—

a violation of the rule of Quantity—to say that I will clean the barn only if

Hazel helps me. For this reason, in many contexts, the use of a sentence of

the form “p only if q” will conversationally imply a commitment to “p if and

only if q.”

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