Uncanny

Mark Liberman's recent posting on the different meanings of can depending on how wide a scope it takes reminds me of my first brush with a scope ambiguity (though at the time I didn't know what it was called). It was back in my sophomore year in high school, in Linda Misenhimer's 6th period biology class. We were having a multiple-choice exam on our genetics unit, and one of the questions went something like this:

John has type AB blood, and Marsha has type B. What blood-type child can they not have?
a. AB
b. B
c. A
d. O

OK, I thought. If John contributes an A gene, and Marsha contributes a B, then their kid will be AB, so that's not the answer. If John contributes an B, and Marsha contributes a B, then their kid will have type B, so that's not the answer. What about type A? Well, John can certainly contribute an A gene, but the rest depends on Martha. If her genotype is BB, then their child will have to have type AB. On the other hand, if her genotype is BO, then she can contribute an O, and their kid will have genotype AO, and type A blood. So item C isn't the answer. That leaves item D, type O. For that to happen, both parents have to contribute an O gene, and since John is AB, that can't happen. Regardless of Marsha's unknown gene, they cannot possibly have a child with type O blood. The answer is D.

I did well on the test, but I noticed that I'd lost two points on that question. After class, I asked why.

"You should have marked both C and D," Mrs. Misenhimer told me.

"But they can have a child with type A, if Marsha's genotype is BO," I said.

"Yes, but if Marsha's genotype as BB, then they cannot have a child with type A."

I didn't understand why this mattered. Until Martha's unknown gene was known, we could not rule out them having a type A kid, so why was I supposed to mark that answer? I asked. Mrs. Misenhimer clarified with her intonation:

"If Marsha's genotype is BB, then they can ... NOT-have a type A child!"

"But ... but ..." was all I could say.

I thought about it some more at home. I came to the realization that the problem was in how we were interpreting the can and the not. Like any other native speaker of English, I was interpreting them to mean, "not able to have," or using parentheses to represent it, NOT(CAN-HAVE). (As semanticists put it, not has wide scope over can.) But it was also possible to interpret them as "able to not have" (or if you don't like split infinitives, "able not to have"); using parentheses, CAN(NOT-HAVE). (In other words, not has narrow scope, and can has wide scope.) A less common interpretation, but sensible enough in context (for example, "Here's what I can do to annoy Jim. I can not laugh when he tells that joke he always tells."). The problem was that Mrs. Misenhimer wanted to have it both ways. She wanted the "not able to have" meaning to say that it was impossible for John and Marsha to have a type O kid, and the "able not to have" meaning to say that it was possible for John and Marsha to have a kid that was not type A. It all seemed so simple now! Geez, using her logic, I should have marked all the options, since it was also possible for John and Marsha NOT to have a type AB kid (if John contributed a B), and NOT to have a type B kid (if John contributed an A).

I went to Mrs. Misenhimer again the next day to point out the ambiguity, so she could see why her question was unfairly graded, and I could reclaim my two points. I wasn't grade-grubbing, you understand; this was an important semantic principle. Unless you're making a pun, you can't just claim every possible meaning of an ambiguous phrase as the one you intended. Imagine someone saying, "I could care less" and meaning both that they didn't care at all AND that they did care!

"I think I see where the confusion came from on this question," I said. "First, you agree that 'not able to' and 'able not to' mean different things, right?"

"No."

I couldn't argue any farther. I had to accept my grade of a 96 on the exam instead of a 98.

UPDATE: I've been thinking about it some more, and I realized that actually it's just barely possible that Mrs. Misenhimer was not trying to interpret the question two ways. The bizarre CAN(NOT-HAVE) interpretation of the question not only allows "type A" as an answer, but also my choice of "type O." That is, if it's impossible for John and Marsha to have a type O child, then it is trivially possible for them to not have a type O child. They CAN(NOT-HAVE) a type O child! But there's still a problem. Not only does the CAN(NOT-HAVE) interpretation allow choices C and D as answers; as I remarked above, it allows every choice as an answer!

So why was Mrs. Misenhimer looking for just answers C and D instead of all of them? The only interpretation I can come up with that requires "type O" and "type A" as answers but not "type AB" or "type B" is this:

What blood-type child is it possible for it to be impossible for John and Marsha to have?

Let's go through the choices one by one. Type AB: it is not possible for it to be impossible for John and Marsha to have a type AB child, since it is a fact that John can contribute an A gene, and Marsha can contribute a B. Type B: it is not possible for it to be impossible for John and Marsha to have a type B child, since it is a fact that John can contribute a B, as can Marsha. Type A: it is possible for it to be impossible for John and Marsha to have a type A child; it is impossible in the case in which Marsha's genotype is BB. Type O: it is impossible for it to be possible for John and Marsha to have a type O child, since it is a fact that John's genotype is AB.

But seriously, that's not how can works, is it? No, the bottom line is this: Mrs. Misenhimer should have given me full credit on that question!