What's ambiguous in: 'ideas are not beyond the capacity of more than 40%'?

Source: Rebecca Gowers. Plain Words (2014 ed). p. 186 Top. I numbered all numerals.

Avoid multiple negatives when you can. Even if you dodge the traps they set and succeed in saying what you mean, you give your reader a puzzle to solve in sorting the negatives out. Indeed it is wise never to make a statement negatively if it could just as well be made positively. A correspondent sends me this:

[1.] The elementary ideas of the calculus are not beyond the capacity of `more than` 40% of our certificate students,

He comments, "I am quite unable to say whether this assertion is that 2/5 or 3/5 of the class could make something of the ideas". If the writer had said that the ideas were within the capacity of at least 60%, all would have been clear.

If I omit modifiers in 1, then 1 becomes 2:

2. The ideas are  not beyond  the capacity of    [some unknown percent] > 40%.

3. Am I correct that not beyond = within? Then 2 becomes:

4. The ideas are  within     the capacity of    [some unknown percent] > 40%.

5. Is 4 correct? If yes, what's ambiguous? Why's the correspondent flummoxed?

• Some people understand calculus, some don't. Of those that don't it can be said that it is beyond their capacity [to understand]. Of the rest, it can be said that calculus is not beyond their capacity. Apparently it is not beyond the capacity of more than 40% of their certificate students- i.e., more than 40% have the capacity to understand the elementary ideas of calculus. – Jim Mar 29 '15 at 4:01
• Without further context, I would read it as 1) 40% of students fail calculus; 2) the speaker thinks this is not due to their inability to understand, but because they are not devoting sufficient effort. In other words, the speaker thinks the students are lazy rather than stupid, and is arguing against "dumbing down" the curriculum by dropping calculus. But the though definitely could have been better expressed! – jamesqf Mar 29 '15 at 5:44

It's not quite so simple as that: not can have ambiguous scope.

You parse it like this:

``````The ideas...are [not beyond] the capacity of [more than 40%] of...students.
The ideas...are   within     the capacity of      [>40%]     of...students.
``````

That is, you take not to qualify only the expression beyond (not beyond = within) and bracket [more than 40%] = [>40%].

But there are alternative parses. Not may be a sentential qualifier:

``````It is not true that [the ideas...are beyond the capacity of more than 40% of students].
``````

And if that is the case, the speaker may rest his denial of the whole on a denial that any particular constituent is true. Which constituent is being denied will only be evident from spoken emphasis and discourse context. Let's take a look at some examples:

A: Prof. B claims that the ideas are beyond the capacity of more than 40% of our students.
B: No, no, no. That's not what I said. The ideas are not beyond the capacity of more than 40% of our students, they're beyond the capacity of more than 40% of our faculty. You lit and history guys don't understand the first thing about mathematics.

A: I'm afraid the ideas are beyond the capacity of more than 40% of our students.
B: It's worse than that. The ideas are not beyond the capacity of more than 40% of our students, they're beyond the capacity of more than 80%.

And this, which I suspect is what the original speaker meant:

A: We need to restructure the curriculum. I'm afraid the ideas are beyond the capacity of almost all our students.
B: That's an appalling exaggeration, A. The ideas are not beyond the capacity of more than 40%. At least 60% have no trouble at all with the ideas.

Note that this is exactly the opposite of your parse. It could be less ambiguously expressed as "The ideas are beyond the capacity of not more than 40%"; but it's very easy for even highly educated writers to overlook possible ambiguities in the heat of debate.

• Thanks. (1) I changed a possible typo: please revert if I'm wrong. (2) In your 1st example, what's the subject of the faculty? I'm inferring that B is a mathematician because she refers to 'you lit and history guys'. But then how can calculus ideas be 'beyond the capacity of more than 40% of our [mathematics] faculty'? – NNOX Apps Jun 2 '18 at 22:28
• @Greek-Area51Proposal Beyond the capacity of more than 40% of our university's faculty, just like "our students" is the university's students, not just the math majors. – StoneyB on hiatus Jun 2 '18 at 22:36
• Thanks again. Are you interested in integrating your comment into your answer, to let us delete our comments? – NNOX Apps Jun 2 '18 at 22:42

Yes, 4 is correct. As to what's tricky about 1; just look at the effort you went through to explain it!

2/5=40%, and 3/5=60%. So the correspondent's difficulty is that it is not easy to tell (without the sort of analysis you just demonstrated) whether the number of students who could handle calculus was at least 40% or at least 60%. You've shown it is the former, but the original form makes that hard to see.