Two out of Five

17 May 2021

Phil Price over at Andrew Gelman’s blog had an interesting post about a certain unnamed principle that caught my attention:

Many years ago I saw an ad for a running shoe (maybe it was Reebok?) that said something like “At the New York Marathon, three of the five fastest runners were wearing our shoes.” I’m sure I’m not the first or last person to have realized that there’s more information there than it seems at first. For one thing, you can be sure that one of those three runners finished fifth: otherwise the ad would have said “three of the four fastest.” Also, it seems almost certain that the two fastest runners were not wearing the shoes, and indeed it probably wasn’t 1-3 or 2-3 either: “The two fastest” and “two of the three fastest” both seem better than “three of the top five.” The principle here is that if you’re trying to make the result sound as impressive as possible, an unintended consequence is that you’re revealing the upper limit.

My first thought was naturally that this sentence is an instance of a scalar implicature. The unintended consequence is a scalar implicature that arises because of a violation of the maxim of quantity. Reebok doesn’t want to say “The third, fourth and fifth fastest runners of the marathon were wearing our shoes” (in the worst case), so they choose to say this instead. We, as Gricean listeners are licensed to derive the hidden inference because of the form of the utterance.

But does Reebok want us to make the inference that one of those three runners finished fifth? Of course not. The construction “three of the five fastest runners” is interesting for this reason – it doesn’t hide the inference, but it weakens it. Consider the alternatives:

  1. Some of the five fastest runners were wearing our shoes.
  2. Three of the fastest runners were wearing our shoes.
  3. Three of the five fastest runners were wearing our shoes.

(3) seems ideal – it’s technically the truth, and it sounds more impressive than (1) and (2). But I can’t shake the feeling that I’m missing something obvious in my thinking. For one, the partitive construction “Three of the five” should lead to a probabilistically stronger inference as Judith Degen has shown in her work. Also (3) sounds more impressive than (2) right? But why is the inference in (3) non-obvious on a first reading of the marketing slogan? Is there an effect due to the usage of the numeral determiners three and five? Assuming the truth of the statement, is Forty of the fifty fastest runners were wearing our shoes as impressive as (3)?

I’ll need to see what work has been done on these constructions, but sentences of this form seem to illustrate Degen’s non-homogeneity assertion: not all scalar implicatures have the same strength.


Andrew Gelman makes an interesting point in reply to my question of why Reebok would use this phrasing when it could lead to a weaker implicature: people know that something is up but we’re still taken in, similar to how we’re more likely to buy something if its ₹99 rather than ₹100.