Devlin's Angle

October 2003

Not-not

SCENARIO A
Sally: "Don't you like it?"

Sam: "No."

So, does Sam like it or not? Most people, myself included, would say he does not. Okay, now consider this variant of the scenario:

SCENARIO B

Sally: "Do you like it?"

Sam: "No."

This time, everyone will agree that Sam does not like it. So what happened to the negation in the first question? The two statements that form the linguistic basis for Sally's questions are:
A: You do not like it.

B: You like it.

Clearly, if A is true, then B is false, and vice versa. Yet when we turn these two statements into questions, they both seem to end up asking exactly the same thing. The effect becomes even more dramatic when you form the questions simply by the way you stress the words (conventionally indicated in writing by the addition of a question mark at the end):
A: You do not like it?

B: You like it?

"So what's the big deal?" you may ask. "Whoever said that natural language should obey the rules of formal logic?" (Actually, as any column writer will know, an endless stream of readers think just that. They jump eagerly to the keyboard to write and point out what they truly believe are grammatical "mistakes", often beginning or ending their lecture with a triumphant "I was surprised to see an experienced writer such as yourself making such a fundamental error." But -- starting a sentence with "But" and insisting that the endless stream of readers I referred to above do "think" and don't "thinks" -- I digress.)

The fact is, the behavior of language (I am deliberately avoiding use of the word "rules" here) is determined by usage. Whatever "Don't you like it?" might have meant in the past, and whatever you think it should mean, today you answer it exactly as you would the question "Do you like it?"

Negation, it seems then, is a tricky concept. Mathematicians traditionally avoid ambiguity by defining negation in terms of truth and falsity. Given any statement A, the negation of A, namely the statement not-A, is declared to be true if and only if A is false. Since questions are viewed as part of mathematical practice, not mathematics, the Sally and Sam puzzle does not have a mathematical analogue. But even with a clear cut definition, things are not so straightforward. Any professor who has given a course on elementary real analysis will have discovered how difficult students find the formally defined notion of negation.

A favorite example of mine is to ask the students to give the negation of the statement:

All American cars are badly made.
The answers I get typically range over:
1. All American cars are well made.

2. All foreign cars are badly made.

3. All foreign cars are well made.

4. Some American cars are well made.

5. Some foreign cars are well made.

6. Some foreign cars are badly made.

7. At least one American car is well made.

8. At least one foreign cars is well made.

9. At least one foreign cars is badly made.

The correct answer is 7. Students who do not see why this is the case are not going to be able to say what it means formally for a real function f to be discontinuous at a point a in its domain, having been told that the formal definition of continuity at a is:
For every positive real number E, there is a positive real number D such that, for all real numbers b a distance less than D from a, the distance from f(b) to f(a) is less than E.
Another example of how everyday language differs from mathematical usage when it comes to negation is double negation. In mathematics,
not-not-A
is traditionally taken to mean exactly the same thing as A. But in English the statement
I don't not like it.

does not say the same as I like it. If it did, more marriages would surely end in divorce and it would be impossible to gently persuade your grandmother not to buy you next year the same hideous bright yellow socks she just handed you this Christmas.

Interestingly, one mathematician won the Fields Medal (mathematics' most prestigious prize) by distinguishing formally between not-not-A and A. Paul Cohen of Stanford showed how to develop a formal logic (called forcing) in which not-not-A and A are different. He capitalized on this distinction to show that certain mathematical statements (among them Cantor's continuum hypothesis) are neither necessarily true nor necessarily false.

What put me in mind of the notoriously problematic nature of negation is when I cast my vote recently in the much-vaunted California recall election. (I'll be out of the country on polling day, so I voted early by mail.) In addition to the recall questions, the ballot included two voter initiative issues. The second of these, Proposition 54, is a constitutional amendment that says (and I quote from the official ballot paper):

Prohibits state and local governments from classifying any person by race, ethnicity, color, or national origin.
The voter is asked to vote YES or NO. If you believe it is a good thing for governments to gather information about race, ethnicity, etc., which box do you fill in, YES or NO? Even in the calm, quiet privacy of my own home, I had to think carefully how to enter my vote. And as a professional mathematical logician, I have made a good living writing and teaching about this kind of thing for over thirty years. My experience teaching negation to many generations of very bright students makes me suspect that many people voting will answer this question by filling in exactly the wrong box. (And remember, in California in particular, for many voters English is not the first language.) Whether those errors will result in a skewed election result is not clear. It may be that there are roughly an equal number of errors in both direction and they effectively cancel each other out. But it is easy to postulate scenarios where the overall effect of the errors results in a different answer to the one that would otherwise have been produced.

It would surely have been far safer to pose the question like this:

Permits state and local governments to classify any person by race, ethnicity, color, or national origin.
Of course, there are obvious reasons why a constitutional amendment, in particular, has to be phrased a certain way. I'm not saying there is anything nefarious going on here. But I do suspect that, whichever way the vote goes on Proposition 54, while we may be able to say, after the votes have been counted, "The people have spoken", I don't think we will ever know for sure what they actually wanted. But then, as the 2000 presidential election count in Florida showed, it's not the case that we don't need to not know how people did not intend to vote. Or have I got this the wrong way round?


Devlin's Angle is updated at the beginning of each month.
Mathematician Keith Devlin ( devlin@csli.stanford.edu) is the Executive Director of the Center for the Study of Language and Information at Stanford University and "The Math Guy" on NPR's Weekend Edition. His most recent book is The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time, published last fall by Basic Books.