## (Not) too P to Q

Peter Howard sent in a listicle at NotAlwaysRight, "10 scams we're not too stupid to fall for", which describes ways that customers will try to fool cashiers, for example by switching price labels:  "it doesn’t take a genius to realise that a $50 bottle of liquor would not be mislabeled as$0.99 cheese-balls in any universe."

Peter observes that the headline "10 scams we're not too stupid to fall for" is not exactly over-negation, in the sense that removing the negation makes things worse rather than better — but still, there's something wrong.

This case is quite similar to the original "No head injury is too trivial to ignore" example — see "No detail too small", 11/27/2009, and "No wug is too dax to be zonged", 11/28/2009. Like may other examples of what we've taken to calling misnegation, such cases illustrate the fact that the interaction of negation and scalar predicates is hard enough for people to analyze that they easily jump to an interpretation that makes sense, even if it isn't the correct compositional analysis of the phrase in question.

Let's start with an analogous example where the sensible interpretation is also the correct one: "(not) too drunk to drive".

The idea of is that there's a scale of drunkenness, and a threshold on that scale such that anyone above that threshold can't or shouldn't drive. If you're "too drunk to drive" then you're above the threshold, and if you're "not too drunk to drive" you're below it.

The opposite of drunk is sober, and so the drunkenness scale is also a sobriety scale. But it doesn't make practical sense to for someone to be "too sober to drive" (though of course there are lots of joke-like things on the internet where this phrase appears…). That's because the scalar direction is reversed — the meaning of the threshold on the sobriety scale is that if you're below the threshold — "not sober enough" — you can't or shouldn't drive. So the evaluative expression becomes  "(not) sober enough to drive"

In simple pseudo-formal language, let's distinguish two cases:

1.  Big P means not Q:   Property P defines a scale with a threshold T, such that if X's P is less than T, Q applies to X; while if X's P is greater than T,  Q doesn't apply to X.
2. Small P means not Q:   Property P defines a scale with a threshold T, such that if X's P is less than T, Q doesn't apply to X; while if X's P is greater than T,  Q applies to X.

The logically correct way to describe these situations then are:

1.  Big P means not Q:   If X's P is less than T, "X is not too P to Q";  if X's P is greater than T,  "X is too P to Q".
2. Small P means not Q:   If X's P is less than T, "X is not P enough to Q";  if X's P is greater than T,  "X is P enough to Q".

Thus we get correct examples like

1.  Big P means not Q:   "X is (not) too drunk to drive"; "X is (not) too proud to ask for help"; "X is (not) too young to study algebra".
2. Small P means not Q:  "X is (not) sober enough to drive"; "X is (not) humble enough to ask for help"; "X is (not) old enough to study algebra".

People sometimes seem to get their directions reversed even in simple cases (e.g. "Although I claim to be an expert on this stuff, I'm not shy enough to ask for help when I need it"). Add some quantificational complexity, and you get things like "No head injury is too trivial to ignore", which are harder to get right and (especially) harder to flag as wrong.

In Peter Howard's example, P is stupid and Q is fall for a simple scam — this is clearly a Small P means not Q situation, and so the way to express the threshold-dependence should be "10 scams we're not stupid enough to fall for". And the quantified relative-clause structure doubtless helped to create and obscure the problem.

1. ### bzfgt said,

April 24, 2016 @ 9:39 am

Somehow until I got to the end I thought you were talking about "it doesn’t take a genius to realise that a $50 bottle of liquor would not be mislabeled as$0.99 cheese-balls in any universe." Your remarks, as you might imagine, didn't always seem to the point. But I convinced myself in the process that this statement is also wrong, or at least infelicitous–if we're imagining an alternate universe in which they are labeled as such, they wouldn't be mislabeled, I would assume, since otherwise the statement is denying that no one in this universe would ever accidentally label a bottle of liquor that way, and in any case the intentional switch is still a mislabeling. Even if we stick to an honest mistake by a clerk, it's hard to imagine that no one in any possible universe could mislabel a bottle of liquor that way…maybe not though, I might have just overthought it since I was thinking about the wrong sentence…

2. ### Andrew (not the same one) said,

April 24, 2016 @ 2:34 pm

Presumably there could be scams we are too stupid to fall for, where the deception turns on getting us to believe something that takes a bit of ingenuity to grasp.

3. ### Gregory Kusnick said,

April 24, 2016 @ 6:47 pm

Andrew: There might be such scams, but how would we know? Being too stupid to fall for them, and way too stupid to see through them, we're presumably too stupid to recognize them as scams.

4. ### Michael Watts said,

April 25, 2016 @ 4:22 am

Here is tvtropes on being too stupid to fall for a scam. Apparently this isn't all that uncommon in stories.

Interestingly, I assumed without reading the article that the intended meaning was "scams we're not too smart to fall for", and that it was a warning.

5. ### David L said,

April 25, 2016 @ 11:27 am

it doesn’t take a genius to realise that a $50 bottle of liquor would not be mislabeled as$0.99 cheese-balls in any universe

Cosmologists who are partial to the multiverse hypothesis argue that the number of universes parallel to our own could be as high as 10**500 — a very big number. If that's so, it's far from implausible that there could be at least one universe in which strong liquor is as cheap as water and cheese balls are rare and expensive delicacies.

6. ### Terry Hunt said,

April 25, 2016 @ 12:07 pm

@ David L

A space habitat in which water and the other materials necessary to produce "strong liquor" were available, but dairy products (absent cows) were not, seems to me an entirely plausible near-ish future scenario in this universe. However, we perhaps stray from the point.

7. ### Jerry Friedman said,

April 25, 2016 @ 12:42 pm

Is there a good linguistic term for words with opposites, such as "stupid", "drunk", "trivial", and "ignore", which show up so often in these erroneous sentences?

bzfgt: Yes, I think you may be overthinking a rhetorical exaggeration. However, I agree that the "it doesn't take a genius" sentence is the kind that can easily go wrong.

8. ### Brett said,

April 25, 2016 @ 3:58 pm

@David L.: That 10^500 is not the number of parallel universes but a crude estimate of the number of possible compactifications of the extra dimensions in superstring theory to make the existenct (3+1)-dimensional vacuum.

As I pointed out in a comment thread here years ago, when the New York times wrote an article about this, their Web site rendered the number as "10500," since their style sheets had not implement superscript fonts.

9. ### January First-of-May said,

April 26, 2016 @ 12:31 am

In any case, 10^500 is a very small estimate for the number of parallel universes (despite being, by most people's standards, a very big number), and very possibly wouldn't have been enough for such unusual things to happen (though it's more a question of said universes' size and diversity).

The figures I've read for the number of parallel universes were more like 10^10^120 (which means 10^10000….000 with 120 zeroes in the exponent).