As recently noted, people often get confused about English phrases involving negatives combined with other negatives, modals, or scalar predicates, and there's a series of Language Log posts that collectively offer several (non-exclusive) hypotheses for why this confusion is so easy to fail to miss:
- Our poor monkey brains just can't deal with complex combinations of certain logical operators;
- The connection between English and modal logic may involve some unexpected ambiguities;
- Negative concord is alive and well in English (or in UG);
- Odd things become idioms or at least verbal habits ("could care less"; "fail to miss"; "still unpacked").
Yesterday's post specifically involved expressions like "cannot underestimate X" or "X cannot be underestimated", as a way of saying that "X is very large or important"; and I followed Lila Gleitman and many others in assuming that these phrases are examples of the class of common logical errors involving negation, modality, and scalar predication. We seem to be saying that X is so large that it's impossible for us to underestimate it — that is, our estimate of X will always be greater than X, no matter how large our estimate is. And this will be true if X is rather small, which is not what we're trying to say.
Now, when I say "common logical errors" I mean "things that people say or write a lot, which mean more or less the opposite of what their authors want to convey". Thus "cannot underestimate" gets 55,000 Google hits, and "can't underestimate" gets 284,000, and it looks like pretty much all of them have the meaning that I just claimed is a logical solecism. Similarly, the various forms of "fail to miss" get almost half a million hits.
That doesn't mean that I'm wrong to call these phrases mistakes — there are a lot of web hits for common misspellings. Still, I got a few protests in the comments from sensible people who refuse to be persuaded that "cannot underestimate" is logically backwards, and so I thought I'd take a closer look.
I'll start by reconstructing (what I think is) their argument, which is similar to one that Barbara Wallraff made a few years ago.
Among the many meanings of the English modal can is one that the OED glosses "to be allowed to, to be given permission to" (OED sense 6.b.), and the AHD glosses "used to indicate that which is permitted" (AHD sense 2b.). This sense is often viewed with alarm by those who feel that it's a mistaken colloquial substitution for may. Thus the AHD entry has a usage note that begins
Generations of grammarians and teachers have insisted that can should be used only to express the capacity to do something, and that may must be used to express permission.
Nevertheless, this sense is common even in formal writing, especially in negatives. For example, in Volume 4 of In Conversation with God, written by an Opus Dei priest named Francis Fernandez-Carvajal, and translated and published "With ecclesiastical approval", we're told that
In the way we dress, in personal hygience, when playing games, we cannot be careless about decorum and modesty. We cannot be seen in a place that is unbecoming for a good Christian, even though it is the fashion or most of our friends go there.
The author and translator clearly do not mean to tell us that we lack the capacity to be careless about decorum and modesty, or to be seen in an unbecoming place — rather, their point is precisely that we do have the capacity to do these things, but we are not permitted, by religious principles, to exercise that capacity.
Now, it's a theorem of deontic logic that if it's not permissible that A, then it's obligatory that not A; or in symbols
¬PA → O¬A
This follows straightforwardly from the fact that PA ("A is permissible") is defined as ¬O¬A ("not obligatory that not A"), and ¬¬O¬A becomes O¬A by cancellation of the double negative.
And since "cannot" can mean "not be permitted to", while "must not" or "should not" can mean "be obliged not to", it somewhat confusingly follows that "cannot" sometimes means the same thing as "must not" or "should not".
That is exactly true in the quote from Fernandez-Carvajal, where "We cannot be seen in a place that is unbecoming for a good Christian" and "We must not be seen in a place that is unbecoming for a good Christian" and "We should not be seen in a place that is unbecoming for a good Christian" all can be interpreted to mean essentially the same thing. Similarly, "We cannot be careless about decorum and modesty" is the same as "We must/should not be careless about decorum and modesty".
Web search suggests that the particular phrase "cannot be careless", meaning "must not be careless" or "should not be careless", is a favorite with religious authors:
We have to differentiate between the singular and the plural form of a word; we cannot be careless about it. Sin and sins are different in the original language.
…Allah (SWT) means we should keep our defence ready, we cannot be careless about the defence of the Muslim Ummah…
Since we are told to spend much time in prayer, we cannot be careless and must strive to pray only as the Lord would have us to pray.
We cannot be careless, thinking that we can say things rashly. [...] We have to fear God and must not speak in a rash way.
To know the word of God we must study the bible carefully: we cannot be careless when handling the word of God.
In a similar vein, we find this in the translation of one of Fidel Castro's speeches:
It is important that at the conclusion of the harvest we can say we have fulfilled our sugar production plans. That is why we cannot fail anywhere. We cannot be careless in any province. No sugarmill can be careless, and anytime a problem emerges, we must have a solution.
Again, Castro is not saying that he and his hearers do not have the capacity to fail or to be careless, but rather that it is not permitted (by the principles of communist rather than catholic ethics) for them to do so.
Similarly, when someone writes that
We cannot be careless about our diets and assume that all is well with our bodies because we take every known vitamin.
he is not claiming that being careless about diets is beyond our capacity, but rather that there is an ethical system, a system of permissions and obligations, that forbids us to do it. Here the ethical foundation is neither religious nor political, but rather scientific — the source is Donald Gray Cook, The Science Book of Wonder Drugs, and it's something like the laws of biochemistry, rather than the word of God or the Communist Party line, that determines what is permitted or obligatory.
This strange equivalence of "cannot" and "must not" does not generalize to the cases where can is used to mean "to have the power, ability or capacity". Thus the proverb "An empty sack cannot stand upright" doesn't mean that "An empty sack must not stand upright". And it's true that I can't hit a major league curve ball, but I refuse to believe that I shouldn't.
Nor does the can that is used to "express a possible contingency" generally allow the "cannot" = "must not" equivalence, since again, we don't use must to express the right kind of necessity. Thus it turns out to be true that an + bn = cn cannot be solved for integer n greater than 2 and non-zero integers a, b, c — but it would be quite odd to say that this equation must not be solved.
OK, let's go back to "cannot underestimate". Consider a random example from the web:
You cannot underestimate the importance of link building to boost your website in the search rankings.
If this means that link building is so important that you do not have the capacity to underestimate it, the author has clearly said the opposite of what he wants to say. Likewise, if it means that link building is so important that it's not logically possible for you to underestimate it, he's again gotten things backwards.
On either of these construals, we're in the territory of hypothesis (1) in my list: our poor monkey brains just can't reliably cope with this combination of negation, modality, and scaling. If a particular expression of this kind becomes frequent enough, we might move into the area of hypothesis (4), since idioms need not be semantically compositional.
But could the correct explanation be hypothesis (2)? Can the subtlety of modal logic rescue "cannot underestimate"?
For that idea to work, this "cannot" must involve the deontic can, the one that means "is permitted to". And my reaction has always been that this is preposterous — in our chosen example, surely the author doesn't mean to tell us that we're not permitted to underestimate the importance of link building.
Or does he? As the examples about sugar cane and vitamins show, people sometimes talk about political and scientific systems as if they were ethical ones. Perhaps in this case, the author is, in effect, preaching to us about search engine optimization — treating it as as a system of permissions and obligations, in which what we cannot underestimate is therefore also what we must not underestimate.
I'm not philosophically or temperamentally inclined to treat estimating the importance of link building (or, in the original example, of grassroots organizing) as a matter of morals. But your mileage, as they say, may vary.
This explanation also sheds new light on the real-world examples that Lila Gleitman sent me last year:
(1) The importance of this position cannot be underestimated.
(2) The importance of this effect shouldn't be overestimated.
(1) came from one of Lila's papers, and her friend Barbara Landau teased her about this logical mistake. (2) came from one of Barbara's papers, and was the basis of Lila's revenge. But in both cases, the context suggests that the authors were preaching to their readers about the positions and the effects in question; and as a result, perhaps these weren't mistakes after all.