"Cannot underestimate" = "must not underestimate"?
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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.
Tim Silverman said,
November 6, 2008 @ 8:52 am
I think speaking of these deontics as "moral" narrows their meaning too much. In any activity directed to some sort of goal, there are some actions that are necessary or helpful for attaining that goal, and others that make it hard or impossible to attain. It is quite normal to speak of these in deontic terms, either through a simple imperative ("add 1 tbsp of sugar") or using modals of permission ("you can add the lemon juice at this point") or obligation ("you must ensure that the potassium cyanide has completely dissolved before serving"). And similarly with negative instructions or injunctions.
Thus we see here goals like a successful harvest, a productive economy, health, correct understanding, a tasty meal, an undetected assassination, etc …
[(myl) Very well put. And I agree, to a point.
But are ethically-interpreted deontic statements merely an instance of a more basic logic of goal-directed activities? Or are the wider goal-directed uses metaphorical extensions of a basic logic of morality? Or is there some basically different sort of modal logic involved?
It won't work to interpret all deontic statements as statements about goal-directed activity — law codes, bureaucratic regulations, etc. need not have any coherent goal, especially from the point of view of those who are simply compelled to obey them, no matter how arbitrary they may seem. So either examples like yours are — as I suggested in the body of the post — cases where the contingencies of cooking or poisoning are being treated metaphorically as ethical systems, or else there's another sort of (non-deontic) modal logic, having to do with goal-directed activity, whose operators are also possible senses for English can and must. ]
Mark P said,
November 6, 2008 @ 9:09 am
I know immediately what someone means by "I couldn't care less," but I have to think a little when someone says that something "cannot be underestimated." It might be possible to justify this usage but it seems easier to avoid it in the first place.
mgh said,
November 6, 2008 @ 9:25 am
"cannot underestimate" => ~55,500 google hits
"must not underestimate" => ~55,800 google hits
it seems reasonable to think these are used interchangeably
[(myl) They are indeed used more or less interchangeably, but I don't think that the similarity in their hit counts is a very strong argument for that conclusion.]
mgh said,
November 6, 2008 @ 9:29 am
as an afterthought:
"should not underestimate" => ~242,000 hits
"dare not…" => ~2,840
"do not…" => ~564,000
"better not…" => ~5,630
what's notable is that these hits are mainly in the form of advice, not consistent with the interpretation of "cannot" as "being unable to" (in fact, "unable to underestimate" gets exactly 2 hits).
Jonathan Mayhew said,
November 6, 2008 @ 9:48 am
I wrote a letter of recommendation in in which I wrote
"I cannot say enough good things about her."
In a later conversation:
"Did you write your recommendation for her."
"Yes, but unfortunately I couldn't say enough good things about her."
Andrew said,
November 6, 2008 @ 9:54 am
Calling "cannot underestimate" a logical mistake seems to me in error. Not that it is logical; but surely it is illogical for effect? I see a gradation:
should not underestimate: weak obligation
must not underestimate: strong obligation
cannot underestimate: stronger obligation, with hyperbole
If this is the case, then your hypothesis (4) would be correct: that a hyperbolical usage ends up becoming common usage.
kip said,
November 6, 2008 @ 11:00 am
"Cannot" meaning "must not" seems to me to be common when giving advice/instructions. Like "you can't listen to what ignorant people tell you."
Maybe there is an implied second clause which would make the "unable to" meaning work? Such as "you can't X [or Y will happen]" or "you can't X [if you don't want Y to happen]" or "you can't X [because you don't want Y to happen]". That certainly seems to be the case in the religious advice examples: "We cannot be seen in a place that is unbecoming for a good Christian [or we will no longer be seen as good Christians]". In this case, if it is a presupposed fact that we want to be seen as good Christians, then the implication is that we are unable to be seen in an unbecoming place.
Of course, I don't think the "cannot underestimate" example falls under this case, except perhaps if it is spreading as an idiom (4).
Rachael said,
November 6, 2008 @ 11:04 am
I agree the phrases are ambiguous, but I'm perfectly happy with the deontic explanation rather than the logical-error one. Suppose the website author had written "You cannot neglect link-building", and the Obama campaigner had written "This grassroots movement can never be ignored". These are clearly deontic (or the second one might even be a "can" of ability, as in "we've had so much effect that it's no longer possible to ignore us"), and make logical sense. But "neglect" and "ignore" are fairly similar in meaning to "underestimate", so I think the sentences still make logical sense with "underestimate" substituted back in.
A similar example:
"I was afraid you'd think less of me for doing that."
(lovingly) "Don't be silly, I couldn't possibly think less of you."
One interpretation is "I already think so little of you that it's not logically possible for me to think any less", like the Mad Hatter telling Alice she couldn't have less tea. But the intended meaning was "I couldn't possibly stop holding you in the high regard I do now." The sentence is genuinely ambiguous; although the first interpretation is closer to a strict translation into formal logic, there's nothing *illogical* about the second.
Alexis said,
November 6, 2008 @ 11:57 am
Thanks for exploring this further. The 'link building' sentence seems more straightforwardly wrong to me than the original "cannot be underestimated", so it helped me see what was going on, as well as the explanations of what senses of 'can' can be used as 'must'.
I think one trouble is that the reading you're claiming is so clearly, in most cases, not what the author intended that people assume, in good Gricean style, that the person must have meant something more sensible (that their intelligence should not be underestimated, one could say ;)
And after quite a bit of experience hearing these overnegations, we stop seeing them as 'wrong' because they are common and can easily be construed properly, and thus assume they must not really be mistakes. I really couldn't see at first what was wrong with "cannot be underestimated". It was only when I thought about what the phrase means on its own, and saw other contexts, that I realized it had the opposite meaning from the author's intentions.
lemuel pitkin said,
November 6, 2008 @ 1:11 pm
Doesn't this usage often include an implicit "if we want to achieve our goal"? Then the cannot/must not equivalency makes perfect sense. As in the Castro speech, where the implicit clause follows directly from the first sentence of the excerpt. If the sugar prduction plan has quotas for each mill, then it is literally true that we cannot miss our target anywhere and still fulfill the plan. No logical error there.
Mark Young said,
November 6, 2008 @ 1:59 pm
What I find confusing is other people's confusion on this topic. "Can't" has meant something very much like "must not" and "should not" (with stronger emotional content) for as long as I've been speaking English. Sure, the teachers tried to train us out of it (especially when used as a positive instead of as a negative — "Can I?" instead of "May I?"), but that only shows that that was the way we used it. (Same with singular "they".)
[(myl) I'm afraid that you are, in fact, rather confused. This has nothing to do with the "can" vs. "may" issue. The point is that in certain(negative) cases, "can" (or equivalently, "may") seems to mean the same thing as "must" or "should". This is slightly surprising, since their basic meanings are quite different. And the fact that "can" in the sense of "be permitted" only dates to the 19th century is no more relevant than the fact that the original meaning of "can" was "know", a sense in which it was used up to the 17th century.]
Are there places in the English-speaking world where children do not use "can" instead of "may"? Are there places where "You can't!" would not be recognized as a claim very much like "You mustn't"?
[(myl) Everywhere, as far as I know, these are often very different — what child would ever take "I bet you can't do it!" to mean "I bet you mustn't do it!"? If you'll read the post above, you'll see an attempt at an explanation of why these are sometimes the same and sometimes different. ]
Mark Young said,
November 6, 2008 @ 2:03 pm
Oh, a conjecture popped into my head — "You can't do X" is like "You mustn't do X", but the former is an expression of a general rule, while the latter is directed at a particular audience. Hence "You cannot underestimate the X" is equivalent not to "You guys shouldn't underestimate the X" but to "People shouldn't underestimate the X".
Prolly someone said that already….
asdf said,
November 6, 2008 @ 2:43 pm
I wish I could remember this better, but there was a sketch on tv (snl?) in which the chief engineer(?) of a nuclear power plant(?) retires, telling his coworkers "You can't have too much coolant" (something along those lines). Later, half the coworkers argue that he meant "You can put in as much coolant as you like, no limit" and the other half "You must not put in too much coolant." The sketch ends with a mushroom cloud.
Coby Lubliner said,
November 6, 2008 @ 3:40 pm
I wonder if some of the "cannot"s occurring in the translations from Fernandez or Castro are mistranslations of the Spanish no se puede, which can mean "one must not" just as well as "one cannot".
All in all I tend to agree with Mark's opinion that "cannot underestimate" is a simple case of confusion over cumulative negations. "Misunderestimate" is a famous example.
Lance said,
November 6, 2008 @ 4:31 pm
I've studied modals a fair amount, and this is still dizzying to me.
The empirical fact, setting aside the overnegation for a moment, is that "You can't" is, roughly speaking, interchangeable with "Don't": "We cannot be careless" is the same as "Don't be careless" (allowing for the small variation in 1st-vs-2nd-person). Again, of course, that's only true on a deontic reading: "You can't get there from here" doesn't mean "Don't get there from here."
So "You can't underestimate", with its odd overnegation, means the same as "Don't underestimate", which has no overnegation at all, even though all the same negative morphemes are there. That's already enough to make me suspect that this really is just idiomatic.
But I'm also not sure that it's not compositional. You might tell a soccer goalie
and that's false on both the epistemic ("It's not possible for you to stay in one place"—sure it is; I spent most of my soccer-playing childhood staying in one place) and the deontic ("It's not permitted for you to stay in one place"—again, I was never removed by a referee for breaking the rules of the game) readings. But as Lemuel says above, there's something implicit here. What the above advice means is actually something like
And that is true on the epistemic reading: there are no (nearby) possible worlds in which both the goalie stays in one place and his team wins the game.
Is that a reasonable thing to elide? Not in a literal "it was present at deep structure and deleted" syntactic sense, but building that into the semantics may not be so bad. Perhaps this is neither epistemic or deontic, but anankastic (i.e., von Fintel and Iatridou's analysis of "You must take the A train if you want to go to Harlem")?
Rubrick said,
November 6, 2008 @ 6:15 pm
"And it's true that I can't hit a major league curve ball, but I refuse to believe that I shouldn't."
I really think you shouldn't. Running onto the field of a major league baseball game, bat in hand, is likely to be viewed very unkindly.
FWIW, my guess is that in the vast majority of cases, "can't be underestimated" is in fact an error, and the speaker really meant "can't be overestimated". I'm at least sure that if I said that, that would be why. However, I suspect that part of the reason for the error is mental pull from cases where "can't underestimate" would be correct in the sense noted above, as for example "We can't underestimate the power of Obama's ground operation [if we hope to defeat him]."
Also, I think "underestimate" simply pops out of people's word-stores more readily than "overestimate" (it's about four times as common on Google).
Douglas Dee said,
November 6, 2008 @ 7:35 pm
I think that Lance's soccer goalie example illustrates the key point.
I've never found these "cannot underestimate" examples problematic, because it seems to me that they are part of a large class of sentences in which people say "X can't do Y", while meaning "X can't do Y without suffering unacceptable adverse consequences."
That is, the example, (1) "You cannot underestimate the importance of link building to boost your website in the search rankings."
is a shorthand for "You cannot underestimate the importance of link building . . . [without crippling your efforts to improve the site's rankings]."
It seems to me that the same explanation covers many examples that might be taken to be "deontic can":
(2) "I've had it. I'm going to walk in there and tell the boss what a big fat jerk he is!" — "You can't do that! [without getting fired and then regretting your outburst]."
The point in (2) is not so much that there's an ethical obligation not to say that to the boss, but that the consequences would be undesirable.
Also consider,
(3) "You cannot ignore the importance of link building to boost your website in the search rankings."
This has "ignore" rather than "underestimate", but seems to me to mean essentially the same thing. If (1) is a "mistake" in the use of "underestimate", then is (3) a "mistake" in the use of "ignore"? That seems implausible. It seems more straightforward to interpret them as "You cannot underestimate/ignore/neglect the importance of link building [without having bad things happen]."
David Schwartz said,
November 6, 2008 @ 11:43 pm
It's quite obvious that phrases like "we cannot fail" can mean either that it is not possible for us to fail because our victory is certain or that our victory is so important that we must do everything we can to avoid failure. It's almost always clear from context which is meant.
Someone who says "you cannot underestimate the importance of a college education", unless they're being silly, means that you are at risk of doing so and that the consequences of allowing yourself to do so are dire. This usage is simply idiomatic. It doesn't follow from any complex logic.
JanetK said,
November 7, 2008 @ 5:51 am
I would have never seen a difference between 'cannot underestimate' and 'must not underestimate' other then one of emphasis. I only ever use 'may' for emphasis; it is a weird, awkward, uncomfortable, little word. It is a word that is written often but spoken rarely. Further, I am surprised by the whole discussion and therefore I have probably been saying ambiguous things about over and under estimating all my life, leaving a trail of confusion behind me – oh my god. I'm nearly 70 so it is probably too late to change either.
gordonoz said,
November 7, 2008 @ 7:38 am
What about `You can't be serious!'? That doesn't mean that you are not allowed to be serious, but that it is impossible for a reasonable person to say what you are saying and be considered to be serious.
Douglas Dee said,
November 7, 2008 @ 8:35 am
@ David Schwartz:
I think we all agree that "we cannot fail" has the two readings you describe — one "possibility" reading that "it is not possible for us to fail because our victory is certain" and one "obligation" reading that "our victory is so important that we must do everything we can to avoid failure."
Likewise with your college example, except that here (as you say) the "possibility" reading is pragmatically silly, and no one would use it except as a joke, so the obligation reading is the obvious one.
The question, though, is whether the same explanations apply to the "link building" example. Mark Liberman (if I'm not misinterpreting him) is inclined to doubt that, because in his view the obligation reading involves (at least metaphorically) treating the sitation as imposing an ethical obligation and he is "not philosophically or temperamentally inclined to treat estimating the importance of link building . . . as a matter of morals. "
The "cannot fail" example can easily be looked at as a moral question. The "college education" example? Maybe. The "link building" example? Not likely, at least for Mark Liberman.
My own view is that all of them involve not an "obligation" reading but reading of "possible (without unacceptable consequences)."
That is, while it's possible for us to fail, and underestimate the importance of education and link building, it's not possible to do so without unpleasant consequences.
Mark Liberman said,
November 7, 2008 @ 10:55 am
David Schwartz: This usage is simply idiomatic. It doesn't follow from any complex logic.
If you mean that "cannot underestimate" is an idiom, then I don't think you're paying attention to the discussion, since the sense in question is available in many other uses of cannot (and related things happen to other modals as well).
If you mean that these uses of cannot are idiomatic, this is not helpful, since the point of the discussion is to determine where such examples work (i.e. "are idiomatic") and where they don't. The "complex logic" arises from the attempt to be explicit about this.
If you simply mean that you're happy that you share with other fluent speakers of English the ability to understand such examples intuitively without analyzing them, then you're reading the wrong weblog.
Killer said,
November 7, 2008 @ 6:17 pm
I would put "cannot underestimate" in the same class as "each one was better than the next" (when of course people mean "each one was better than the last"). Which is to say: wrong — but unlike "better than the next," it's possible to make logical case for the incorrect version. Surely most people use the phrase with the intention of expressing "cannot overestimate," but they get it wrong simply because they've been hearing other people saying it that way. Like "could care less." Which is probably just fine … but I can't help feeling that there's a lot of rationalizing in these comments that's driven more by what is logically arguable than by what people probably mean.
dr pepper said,
November 7, 2008 @ 11:33 pm
@ asdf
You beat me to the reference.
That was Ed Asner on SNL. He played the chief engineer who told his employees: "just remember: you can't put too much water in a nuclear power plant", just before going on vacation. And after he leaves, there is a malfunction, and the plant starts overheating. And while the engineers are still arguing over the meaning of his statement, the plant goes critical.
Ben Ostrowsky said,
November 16, 2008 @ 2:04 am
My grandfather recounted a performance review he'd received at work. It included the statement "I cannot praise this man too highly." He would, according to family tradition, ask rhetorically, "What the hell does that mean? Sure you can! Go right ahead!"
Jakob said,
December 5, 2008 @ 1:49 pm
Maybe the Castro and link-building cased involve an implicit conditional. If we want to achieve our goals then we cannot do such-and-such. If you want a lot of page hits, you cannot underestimate the importance of link building.
capt_crea said,
January 2, 2009 @ 1:38 am
As a Spanish speaker for the rest of my life, including my previous life, I dream to reach the day when I understand the real o modal o whatever difference between "must not" and "cannot". The quotes shown here are from Spanish speakers, like me. There are communists, catholics and more people speaking Spanish in some areas of the world but, I afraid, they are not saying the same when they use "no puedo" or "no debo" (implied in "no podemos" or "no debemos" as a kind of pseudamayestatic antiplural usual among dictators and other supreme authorities). They are not saying the same thing as any logic statement.
I am happy to see that the American democracy has elected a "We can" to lead the new-whatever in the world. This could be the solution to many linguistic questions. The Spanish "podemos" never looked so credible as now when it translates young and happilly the new "We can".
Submitter said,
June 10, 2009 @ 6:40 pm
I must admit that I get puzzled about this equality "Cannot underestimate" does it really equal "must not underestimate"? I am not native in english, but why don't we just agree we could just say "do estimate".
Mark, your case is very convincing but for me both sound right. When I have this feeling I usually go for the alternative. (negative 1) multiplied by (negative something) (-1)x(-A) = +A.
So my solution is that instead of saying "cannot underestimate" , or "must not underestimate" we should all say do estimate or plain estimate. And yes, I am well aware that by doing this substitution I fit perfectly in all the first three categories that you described in the beginning of this post.
John said,
March 23, 2011 @ 11:22 am
There's a deontic meaning of "can" that approximates to "be able to do X and still acheive a desired result", or in short "be able to afford to do X".
For example, "Can I use brown sugar in this recipe?" Obviously it's physically possible for you to mix brown sugar in with the other ingredients. If it's your kitchen, then you certainly have permission to do so. Unless you follow a seriously restrictive religion or dietary regime, you aren't morally obligated not to. The question here is "If I use brown sugar, will the recipe still work satisfactorily?" Compare sentences like "You can't read a book and watch TV at the same time".
By my interpretation, examples like the search engine optimization one are using that kind of "can". "You can't underestimate these factors" could thus be seen as an abbreviation of "You can't (run a successful business if you) underestimate these factors", or "You can't (afford to) underestimate these factors".
John said,
May 11, 2014 @ 1:11 pm
Just for comparison's sake, one could say "You cannot afford to underestimate…".
I imagine the fact that there are lots of structurally-similar ways of phrasing this kind of statement might make people slip up and confuse/combine them.
"You cannot afford to underestimate X"
"You must not/should not underestimate X"
"You cannot OVERestimate X".