[Below is a guest post by Gabriel Dupre]
Sentences of the form “No X is too Y to Z” are, in many cases, nightmares to process. The interaction of multiple negations (explicit and implicit), scalar adjectives and modals makes correctly interpreting such sentences very difficult. This has long been noted by linguists and psychologists. However, all of the accounts we can find of these types of sentences not only note the difficulty of a first-pass parse of the sentence, but also misinterpret the literal meaning.
The standard example of this kind of sentence is “No head injury is too trivial too ignore,” allegedly found on the wall of a triage department at a hospital. The usual, unreflective interpretation is as of the sentence “No head injury is trivial enough to ignore.” One hears this sentence as denying that there is some boundary along the ranking of head injuries from trivial to serious below which a head injury may responsibly be ignored. That is, no matter how trivial a head injury is, one should not ignore it.
The standardly proposed ‘literal meaning’ may be analogized to the more easily interpreted sentence “no basketball player is too short to dunk.” That is, the sentence means that no matter how trivial a head injury is, it can be ignored.
It is usually stated, then, that it follows from this literal interpretation that every head injury may be ignored. (I.e. it is claimed to follow from “No X is too Y to Z” that “Every X may be Z’d.”)
This analysis is seen in Wason and Reich’s original paper on this topic [Peter Wason and Shuli Reich, "A Verbal Illusion", The Quarterly Journal of Experimental Psychology , 1979]: “However, the injunction, Ignore all head injuries, which follows from the sentence, No head injury is too serious to be ignored, is unusual if not actually incredible,” and in a 2009 Language-log post by Mark Liberman ["No wug is too dax to be zonged", 11/28/2009]: “Against this background, we assert that T is in fact so far along the scale of smallness that there are no missiles "too small to be banned", i.e. on the not-banned side of T:
|← less small||–||more small →|
It follows that all missiles should be banned.”
In conversations with Levi Roth and Bill Kowalsky, we convinced ourselves that this inference is incorrect.
There is an uninteresting way in which a counter-example to this inference can be seen: even if no head injury is too trivial to ignore, it could be that a head injury may be ignored in virtue of some other factor. For example, if an evil man with a possibility of becoming president had a head injury, it may be morally required to ignore it, for the sake of the country. What is interesting is that it does not even follow from this sentence that no head injury can be ignored simply in virtue of its location on the trivial-serious scale. To demonstrate this, we will build up this sentence from its parts, demonstrating what each contributes to the meaning of the sentence, and then provide a counter-example to the alleged inference.
Take first the complex predicate “too serious to ignore”. The extension of this predicate presupposes a scale of seriousness along which we can plot entities from most to least serious. For this predicate to have an extension, there must be some point along this spectrum, such that everything above that point cannot be ignored.
Replacing the high-end-of-the-scale predicate ‘serious’ with the low-end-of-the-scale predicate ‘trivial’, we get the complex predicate ‘too trivial to ignore’, the extension of which will be everything below a certain point on the trivial-serious scale. Everything in this extension cannot be ignored.
Now take the sentence “this head injury is too trivial to ignore.” This places a particular predicate within the previously described extension, and says of it that, in virtue of its extreme triviality, it cannot be ignored.
The sentence “No head injury is too trivial to ignore” then, states that there are no head injuries in the extension of this predicate. That is, this predicate does not provide a partition of head injuries along this scale. Every head injury falls above the boundary provided by this predicate, and so no head injury is such that it cannot be ignored in virtue of its triviality. No matter how trivial a head injury is, it may still be ignored.
However, that this predicate does not partition head injuries does not mean that no predicate does; not even that no predicate on this same scale does. That there is no boundary below which no head injury can be ignored, it does not follow that every head injury can be ignored. There may well be a boundary along this scale such that no head injury above this boundary may be ignored. Thus, there is no contradiction in the claim “No head injury is too trivial to ignore, but some head injuries are too serious to ignore.” But, the latter clause of this sentence explicitly denies the standardly perceived entailment of the former: that every head injury may be ignored.
The interest of all of this is that even with careful reflection from the people most attuned to these kinds of parsing difficulty, it has (from what I can tell) gone unnoticed for decades that the entailments of these kinds of sentences are not what have been claimed of them. An extra level of interest comes from the fact that the original, reflective analysis of the sentence (the one that entails the universal claim) appears to display the same resilience that the original, unreflective interpretation (that no head injury is trivial enough to be ignored) does. It is hard, once initiated, not to interpret the sentence as having this entailment, even when convinced that it does not. This seems to be an example of the resilience of learned intuitions.
[The above is a guest post by Gabriel Dupre.]