The Estimation Game

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More than 30 years ago, the famous linguist Mark Aronoff joined Lila Gleitman and others who have gotten under-/over-estimating upside down– "Automobile Semantics", Linguistic Inquiry 1981:

Makes are ranked according to price category. There are three such categories: lower-priced, mid-priced, and luxury. [...]

One of the results of the existence of ranks is that makes can be compared, regardless  of manufacturer. Chevrolet and Ford are thus equivalent, as are Cadillac and Lincoln. Mercury is higher than Chevrolet, etc. The ranks form a value system, and each make can be placed within this system.7

7 The importance of rank in the significance of a make cannot be underestimated. Two phenomena of recent years show this clearly. The first example is that of the Oldsmobile with the Chevrolet engine. Several  years ago, it was discovered that some full-size Oldsmobiles were being sold with Chevrolet engines, without the customer's knowledge. The Chevy and Olds engines were approximately the same, the only major difference being the make. Many of the people who had bought these "hybrids" were furious. There were numerous court cases against GM for the deception, and the affair was finally settled with GM paying millions of dollars in compensation. All along, GM expressed puzzlement, for the interchange of components among different makes is standard practice and is becoming more and more widespread. There was no qualitative difference between the two engines, they said. But the customers and the courts would not buy this. After all, those of us who watched TV in the fifties knew all about being behind the wheel of a Rocket Oldsmobile with its famed Rocket V-8 engine. Chevrolet is OK, but a Chevy is not an Olds, nor is its engine. [emphasis added]

As Tim Zingler has pointed out to me, the context of Aronoff's footnote 7 makes it clear that "the importance of rank in the significance of an [automobile] make" can indeed be underestimated, as demonstrated by the fact that GM did in fact underestimate it.

But I suspect that some readers are still unsure about the logic of cases like this. Google Scholar has more than a thousand hits for "cannot underestimate the importance of" X, and inspection of a sample suggests that they all interpret the expression in the same way that Mark did, to mean that X is of paramount importance.

So let's play the Estimation Game.

We have an oracular Importance Meter, which give us the true (or at least official) importance of anything that we hook it up to, measured on a scale of 0.0 Importance Units to 10.0 Importance Units. In order to play the game, a contestant estimates the importance of something — call it X — and then we connect X to the Importance Meter, and compare the contestant's estimate to the official answer.

There are three possible outcomes:

  1. The contestant's estimate is less than the meter reading, in which case the importance of X has been underestimated;
  2. The contestant's estimate is exactly equal to the meter reading (down to the precision of measurement, which is 0.2 Importance Units), in which case  the importance of X has been correctly estimated;
  3. The contestant's estimate is greater than the meter reading, in which case the importance of X has been overestimated.

Each of the three outcomes is announced with characteristic flashing lights and blaring noises, and there's a board for keeping score of each contestant's mean squared estimation error, and there are prizes, and so on.

OK, now let's imagine a round of the Estimation Game where the CEO of General Motors estimates the importance of tail-light design.  Her guess is 5 Importance Units, right in the middle of the scale. We connect "tail-light design" up to the Importance Meter, and behold, the importance is actually 6:

So the CEO of GM has underestimated the importance of tail-light design.

If she had guessed 7, she would have overestimated

In this situation, it's easy to guess too low or two high. There are 30 possible guesses that are under-estimates (0 to 5.8, incrementing by 0.2), and 20 possible guesses that are over-estimates (6.2 to 10.0 by 0.2), and just one guess (6.0) that's exactly correct.

Now the same contestant is asked to estimate the importance of "rank in the significance of a make". And when we hooked up the Importance Meter to "rank in the significance of a make", we get the maximum possible reading of 10.0:

At this point, it almost doesn't matter what her guess was. All the 50 estimates from 0.0 to 9.8 are under-estimates; one guess is exactly correct (10.0); and there are no possible guesses that are over-estimates. It is literally impossible to over-estimate the importance of X in this case. We literally cannot overestimate it.

Alternatively, if the Importance Meter reads 0.0,  there are 50 guesses that are over-estimates, and one guess that's exactly correct, but it is literally impossible to underestimate the importance of X. We literally cannot underestimate it. The importance of X cannot be underestimated.

So when we say that, as Mark Aronoff did, that "The importance of rank in the significance of a make cannot be underestimated", we (should) mean that no estimate of importance can possibly be too low, in this case, because the Importance Meter's reading is as low as it can possibly be.

But this is clearly the opposite of what Mark meant.

He's not alone — this is one of many cases where the interaction of negation, modality, and scalar predication causes smart people to say or write the opposite of what they mean:

"We cannot/must not understate/overstate", 5/6/2004
"Overstating understatement", 6/22/2004
"Multiplex negatio ferblondiat", 7/14/2007
"Weird logic and Bayesian semantics", 7/15/2007
"'Cannot underestimate' = 'must not underestimate'?", 11/6/2008
"Misunderestimation", 4/4/2009
"Gov. Cuomo and our poor monkey brains", 1/21/2011
"… not understating the threat", 6/5/2012
"(Not) Underestimating the Irish Famine", 9/16/2012
"Overestimating, underestimating, whatever", 1/11/2013
"CIA unable to underestimate the effect of drone war", 4/7/2013
"Misnegation of the week", 5/17/2013
"'Impossible to understate' again", 3/1/2014
"'Hard to understate'", 3/19/2014

This state of affairs should make descriptivists stop and think. Can the authors of a thousand Google Scholar hits — and innumerable other speakers and writers —  be wrong? In this case, I think, the answer is "yes".

But I've argued that "could care less", where modality and scalar predication seem similarly to point in the wrong direction, has simply become an idiom. Shouldn't the same be said for "cannot underestimate the importance"?

I don't think so. As I've argued before, there's a crucial difference.

Whatever is happening with "cannot underestimate" applies equally to "cannot understate", "impossible to underestimate/understate", "hard to underestimate/understate", "difficult to underestimate/understate", "cannot be underestimated/understated", "hard to underrate", "cannot be undervalued", and many other common ways to re-express the same idea.

In contrast, alternative formulations of "could care less" are rare, and can only be understood as bad jokes, to the extent that they're not simply puzzling.  Thus one semantic equivalent to "could not care less" might be "could not possibly have less concern" — and we find this in a published translation of Montaigne's sentence

Si toutefois ma postérité est d'autre appétit, j'aurai bien de quoi me revancher: car ils ne sauraient faire moins de compte de moie que je'en ferai d'eux en ce temps-là.


However, if my descendants have other tastes, I shall have ample means for revenge: for they could not possibly have less concern about me than I shall have about them by that time.

But in this case, Montaigne means to imply that his concern-meter will be pegged at zero, not at its maximum value. And more generally, we don't see things like "I could possibly have less concern" used with the meaning idiomatically assigned to "I could care less". This is the behavior that we expect from an idiom; and the different behavior of "cannot underestimate/understate/underrate/undervalue" is what we expect from a psychologically probable error.

[N.B. The Importance Meter graphics in this post were adapted from the images used in a javascript applet due to Wiesemann & Theis GmbH.]



  1. Brett Reynolds said,

    April 3, 2014 @ 6:24 am

    I think Mark has overlooked one possibility:

    [(myl) I'm well aware of this option, but it requires special equipment, which had not yet been invented in 1981, the year of Automobile Semantics' publication.]

  2. Keith Clarke said,

    April 3, 2014 @ 7:23 am

    I don't suppose this is an original thought, but I don't know how to check.

    Perhaps people who find "can" more idiomatic than "may" when seeking or granting permission are therefore saying "cannot be underestimated" to mean "may/must not be underestimated".

    (anecdote) Growing up in Liverpool, England, in the 1950s we'd often ask "can I go out to play?". A peevish parent might correct that to "may I go to play", but we were understood. This phrase is too long to check with Google ngrams but "can I go out" has risen as "may I go out" has fallen. Which perhaps is evidence that they're not (always) being used in different senses. British English seems to prefer "can I go out" by a margin of 3 to 1 at the moment, somewhat greater than American English.

  3. Scott McClure said,

    April 3, 2014 @ 7:42 am

    I think I'm comfortable with reading 'cannot' as 'must not' or 'should not' in a lot of these 'X cannot underestimate Y' cases. Something like this possibility was discussed in at least one earlier Language Log post ( — the difference here is that the 'X cannot underestimate Y' case isn't about a moral or ethical code so much as it's about conforming to certain principles that are said to make one a savvy guesser.

  4. Ellen K. said,

    April 3, 2014 @ 8:26 am

    Except, Keith Clarke, it also happens with other scalar expressions that don't involve can/cannot. See the paragraph 2 above the blue paragraph near the end of the article. Also, see the links a little above that.

  5. Pflaumbaum said,

    April 3, 2014 @ 10:04 am

    In deciding what is an idiom, should the speaker's own view count for something?

    For instance, if you could convince each under estimator to read this post, I'm pretty sure almost all of them would say, "Oh yeah – I meant can't be overestimated."

    Whereas, judging from internet comments threads, a lot of people who have I could care less in their dialect are not amenable to prescriptive arguments that they're saying the opposite of what they mean. They may not know the NPI argument, or Steven Pinker's sarcasm argument, but they know what feels right.

  6. Keith Clarke said,

    April 3, 2014 @ 10:33 am

    I take your point, Ellen, and I'm probably out of my depth here. Thanks, too, Scott, for the pointer to the 2008 posting and discussion. My monkey brain clearly isn't good at reading lists, because ""'Cannot underestimate' = 'must not underestimate'?", 11/6/2008" is right there in Mark's posting this time.

    Tim Silverman in the first comment to the 2008 posting talks of goal-directed activity rather than moral or ethical choices. There does seem to be a didactic context where adults/teachers/professionals use "can" to mean "should". I will re-read that comment thread.

  7. Kevin Ringeisen said,

    April 3, 2014 @ 2:20 pm

    I've been following these misnegation posts for a while and I'm wondering if phrases like "cannot underestimate" are actually implying the presence of some knowledge about the thing being estimated.

    For example, "it is difficult to underestimate x's importance" should be taken to mean "It is difficult to underestimate x's importance, given the fact everyone has a sense of how important it is."

    In other words, what they mean is "no one *would* underestimate x's importance," while keeping in mind that if no one would do something, the probability of its occurrence would be low, which leads to the conclusion that it would be difficult to find or cause someone who would underestimate x's importance.

    In cases like these I don't think the situation is a blind guess, like with the importance Meter

  8. Kevin Ringeisen said,

    April 3, 2014 @ 2:22 pm

    Ah, accidentally hit submit.

    But rather, there is a very low probability of someone guessing below a six in importance, or whatever the actual number is, and likely a greater probability of guessing too high.

  9. Kevin Ringeisen said,

    April 3, 2014 @ 2:28 pm

    "Hard to underestimate" seems to mean the same thing as "unlikely to low-ball."

  10. Peter Fedak said,

    April 3, 2014 @ 3:24 pm

    The post seemed a little over-the-top to me. I'm skeptical that too many readers were unsure about the logic of your interpretation of these errors in a way that the images would clarify. Most of the comments I have seen on your articles on this topic over the years have been about the modal/deontic/"must not" interpretation, which is not really addressed by elaboration on the "game".

    I must say I am on the same page as Keith Clarke. In this particular example, I understood "cannot" be mean "must not, or else we run into problems, as demonstrated by this anecdote." I am not terribly convinced by an argument that in other seemingly similar cases there is no such possible interpretation, and that the best explanation is one that does not address the meaning of "cannot" that I (and others) seem to be latching onto.

    I see a big difference between "understate" and "underestimate." The latter seems to be about one's mindset, which influences actions, while the former is not. I will grant that, out of your list of the phrases to which your analysis "applies equally," the combination cannot+estimate seems unique in allowing the alternate, non-error interpretation. "Hard/easy" seem explicitly about possibility, and "understate" doesn't make sense when giving advice.

    Perusing that list of ghits you mentioned, I can't say I am dissuaded. The first (as it appears to me,, about carbon sinks, comes at the end of a paragraph about the difference between focusing on sources versus sinks, and the considerations that "cannot be underestimated" are being underestimated by people who continue to focus on sources, say. The second, which gives a 404, contains "Cannot underestimate the importance of good, in-depth assessments" right before "Assessments should not be 'one-off' snapshots of families' behaviours, but should include:", which seems to be admonishing underestimation. In the third, (, "underestimate" occurs in a paragraph which ends "If these steps are not adequately accomplished, good esthetics, comfort, and chewing ability will be missing," again warning against underestimation.

    I would be surprised if these uses were NOT related to the misnegation examples, but, committing the sin of relying on my intuition, I'm hard-pressed to call what I see as an alternate meaning an error.

  11. Brett said,

    April 3, 2014 @ 4:42 pm

    I am persistently amazed by the number of people who always try to defend these phrasings as logical. I can read these these usagess, and most of the time I don't think twice about the intended meaning. However, that doesn't mean that I process the statement as an imperative, or a suggestion about shared prior information. My poor monkey brain usually just doesn't notice the scalar inversion. I see know reason to believe that the authors have not simply made the same mistake.

  12. Rubrick said,

    April 3, 2014 @ 5:09 pm

    I'm with Brett. The determination of some people to justify any usage as sensible cannot be under- — er, over- — eh, screw it.

  13. John said,

    April 3, 2014 @ 5:35 pm


    I understand what you're getting at. But it seems more likely that people are simply making errors. I think Mark's 2004 post "We cannot/must not overstate/understate," linked to above, addresses your point pretty well.

    Also, consider that if your interpretation were true, something would be off about the example under discussion, where the author starts off by stating that the importance of rank "cannot be underestimated," then provides an example of someone who should know better doing exactly that.

    [(myl) I'd add that the deontic-logic explanation, on top of its intrinsic problems, has nothing to say about things like "hard/difficult to underestimate/understate".]

  14. Peter Fedak said,

    April 3, 2014 @ 5:58 pm

    Slogging further through the Google scholar results:

    "Although the majority of strokes that occur during coronary surgery are caused by cerebral macroemboli, we cannot underestimate the importance of microemboli."

    "To be on top as an aviation professional,you cannot underestimate the importance of computers to the industry."

    I don't think it takes very much to defend these usages as logical. To me they make much less sense if interpreted as some exaggerated, erroneous form of "X is very important." There are plenty of indefensible usages, sure, but is it impossible for a correct usage to coexist with an incorrect one?

  15. Kevin Ringeisen said,

    April 3, 2014 @ 6:11 pm

    I agree, my reasoning is less clear when it comes to phrases like cannot, or impossible. "X cannot be underestimated" becomes "No one would underestimate X, and clearly GM, who should know better, does exactly that and underestimates it.

    On the other hand, we say "No one would do X" in all sorts of situations only to find out we were wrong. GM wouldn't have expressed puzzlement otherwise.

    But what I was getting at is, I think, different from what is discussed in the we cannot/must not understate/overstate article. My point was that it doesn't make much sense to assign an equal probability to guessing an importance of 1.2, and to 7.6.

    If I I ask several people how important fuel efficiency is when car-shopping, I expect the answers to cluster somewhere around "important", but if I ask someone "how important is the number of spokes on the wheels?" I expect the answer to be quite close to "not important."

    If we plug those two assertions up to the Oracular Importance Meter, Let's say the former tests at 9.0, and the latter at 0.5.

    But *despite* the fact there are twice as many guesses to the right of 9.0, than to the left of 0.5, I do think it is easier to underestimate the 0.5 than it is to overestimate the 9.0.

  16. Brett said,

    April 3, 2014 @ 9:37 pm

    @Peter Fedak: I agree. Those phrasings seem to be correct. However, my distinct impression is that careful usage like that is not especially common. Most of the time, "cannot underestimate" just seems to be an error. (Interestingly, those two correct, mandative usages were much harder for me to understand than the usual, incorrect ones.)

  17. Viseguy said,

    April 4, 2014 @ 12:07 am

    I hate "could care less"! If it's an idiom, it became such through the mindless repetition of an error (or is it the lazy omission of a slightly hard-to-pronounce syllable?), and well (splutter, splutter) there oughta be a law! I can't swallow the notion that the usage is ironic. Are we seriously to believe that the utterer of "I could care less" really means something like, "(Ask me whether) I could care less"? Try as I might, I can't hear the imperative mood, the defiant request for an interrogatory. And, how, I ask you, is that implied conversation supposed to go?

    "Ask me whether I could care less!"
    "O.K. Could you care less?"
    "So you couldn't care less."
    "That's what I said!"
    "Um, no…."


    "Ask me whether I could care less!"
    "Could you care less?"
    "Yes? You mean no, yes?"
    "'No', as in yes, or 'No' as in no?"
    "So you're really saying you couldn't care less."
    "No, I'm saying I could care less!"
    "Ah! Thank you for your concern."

    You couldn't overestimate how much better I feel now.

  18. Sven Sahle said,

    April 4, 2014 @ 4:32 am

    If I look at the german language (that has many constructs parallel to english) both interpretations of the english term can be translated and both make sense. However, in german the meaning would be unambiguous in all cases I can think of. You could say something like "Bitte unterschätzen Sie nicht die Schwierigkeit dieser Aufgabe" ("Please do not underestimate the difficulty level of this assignment"), e.g. in a teaching context.
    You can also say "Die Bedeutung dieses Gegenstands kann kaum hoch genug eingeschätzt werden" ("It's almost impossible to estimate the importance of this issue high enough"). For me the "almost" part is necessary in the german version, the hyperbole would feel too strong without it.

  19. Keith M Ellis said,

    April 5, 2014 @ 2:15 am

    Surely we need not decide that all such usages have the same explanation? It seems most reasonable to me that some cannots are intended as should nots, a few are intended as unlikely tos, and some are misnegations.

  20. Mark Stephenson said,

    April 5, 2014 @ 11:18 am

    My American wife says "We can't forget to …", by which she means "We really mustn't forget to …" But in my British speech, "We can't forget to …" means "It's impossible for us to forget to …", which is almost the opposite, since now there's no need to make sure we don't forget.

  21. milloum said,

    April 8, 2014 @ 6:39 pm

    Right, so (English-speaking) monkey brains have a difficult time comprehending this mix of scalars, modals, and negatives. That's been established I think.

    In the hope of contributing to the debate, I offer an intuition-based test: try to substitute "underestimate" (or "understate", "undervalue" etc) with "emphasize enough" and see if that sounds equivalent or not.

    "The importance of rank in the significance of a make cannot be underestimated. " => "The importance of rank in the significance of a make cannot be emphasized enough."

    Yep, that does seems pretty much equivalent in context.

    "the importance of X can not be emphasized enough" seems straightforward to this monkey brain: we are talking of something of high importance, and we are certainly not saying that we "should not emphasize it enough", but that it is actually "impossible to emphasize it enough". Not mandative, assertive.

    Now in a clearly mandative sentence like "the importance of X should not be underestimated" our substitution automatically sounds like nonsense, or at least produces a different meaning: "the importance of X should not be emphasized enough", or more grammatically "…should not be emphasized too much". The two phrases are not equivalent in meaning: the original one asks us to be wary of underestimating, the new one recommends underestimating.

    So if we can intuitively, in context, admit that "X can not be underestimated" is equivalent to "X can not be emphasized enough" then we should conclude that the sentence contains a logical error: the intended meaning is not "underestimate" but "overestimate".

    The examples Peter Fedak and Brett identify as correct uses of "cannot underestimate" seem to support my point, as they don't sound equivalent after substitution. thus "can" in these cases actually means "should" or "must".

    One limitation is that estimation is one thing, emphasis is another. But I'm assuming that in most contexts what we're really talking about is in fact emphasis. After all, it's very rare to find ourselves faced with an actual measurable value that is impossible to over- or under-estimate, unless it is, say, the weight of photon (can not be underestimated) or the density of a black hole's singularity (can not be overestimated).

    This points to another possible reason for how fuzzy this underestimation business feels: the fact that we are not really estimating anything in most cases, so that scalars are really a bit inadequate (except if, as Sven Sahle seems to say is mandatory in German, we temper them with "almost" or "hardly"). But I won't go there, this comment is already too long.

    Second (IMO more serious) limitation, I'm aware of the methodological danger of the researcher relying on his own "intuition". I've been thinking about this for a few hours now, so my intuition is pretty much fried.
    Any thoughts?

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