ONLY IF’

Consider the statement:

(1) Your insurance company will PAY up for CDs stolen from your car only if you LOCKED the car door.

Which of the following two statements is this equivalent to?

(2) Y our insurance company will PAY up for CDs stolen from your car if you LOCKED the car door.

(3) Zfyour insurance company PAYS up for CDs stolen from your car, you must have LOCKED the car door. Symbolized: P → L

This question is probably causing you some puzzlement. So far we have assumed that the statement following the ‘if’ is the antecedent, which means that (1) should be symbolized by L → P. But that symbolizes statement (2), which doesn’t seem to say the same thing. One could imagine someone having locked the car door, but left the window open, or forgetting to submit a claim; in which case, even if (1) is true, (2) does not follow.

One way to sort this out is to see what else these statements are equivalent to. Intui­tively, (1) seems equivalent to:

(4) Your insurance company will not PAY up for CDs stolen from your car if you did not LOCK the car door. Symbolized:

Now suppose you locked the car door. It does not follow that your insurance company will pay up—that would be denying the antecedent! But suppose the company pays up. Assuming (4) is true, we can then infer that you must have locked the car door (by Modus Tollens and a couple of Double Negations). But that’s exactly what statement (3) says! So we have (1) is equivalent to (4), and (4) is equivalent to (3). Therefore (1) is equivalent to (3). What we have proved is that “P only if L” is symbolized as P → L. This contrasts with “P if L” which would be the same as “If L then P,” and thus would get symbolized L → P.

Putting the all-important little word ‘only’ in front of an ‘if’ switches antecedent to con­sequent, and vice versa.

Here’s an example involving the little word ‘only.’ Recently Spanish researchers claimed to have found evidence for a black hole circulating around another star in the constellation Scorpius. They found an abundance of the elements oxygen, magnesium, silicon, and sulphur in the spectrum of the visible star. “The only way you can produce an abundance of these elements is through several billions of degrees,” said Rafael Rebolo, co-author of the study and researcher at the Institute of Astrophysics in the Canary Islands. “The only way to reach these temperatures is when a star goes to a supernova situation.” If we use the key

A := an ABUNDANCE of the elements oxygen, magnesium, silicon, and sulphur has been produced. T:= the TEMPERATURE must have been several billion degrees. S := the invisible star went SUPERNOVA.

we can re-express Rebolo’s two statements as A only if T, and T only if S; symbolized A → T and T → S. (The complete argument is given as exercise 3 for this chapter.)

One final note of caution: do not confuse “only if” with ““if only. ”

The following line from a classic pop song:

(5) I’ll be home for CHRISTMAS, if only in my DREAMS.—Kim Gannon and Wal­ter Kent

would NOT be symbolized C → D. In fact, this is not even a conditional statement.

8.1.2