“Vorurteil der Gelehrten. – Es ist ein richtiges Urteil der Gelehrten, daß die Menschen aller Zeiten zu wissen glaubten, was gut und böse, lobens- und tadelnswert sei. Aber es ist ein Vorurteil der Gelehrten, daß wir es jetzt besser wüßten als irgendeine Zeit.”

Nietzsche: Morgenröte. Gedanken über die moralischen Vorurteile (1881), Erstes Buch, 2

 Here are two points (almost) all linguists agree on:

  1. Grammars should not prescribe what language should be like, but describe what language is like.
  2. Grammars should describe a language as an infinite object, to account for linguistic creativity (put differently: there is no fixed number of grammatical sentences).

It has been hardly noticed that these two points contradict each other: from observation, we only know a finite number of linguistic objects; if our grammars describe infinitely many linguistic objects, they are necessarily prescriptive on those objects we do not observe. This means among other that statements such as “natural languages are not context-free” or “natural languages are mildly context-sensitive” are meaningless in the sense that they cannot be falsified empirically – they can only be falsified if we project a finite language into the infinite in a certain way. This however turns out to be a highly non-trivial issue!

This problem comes with a number of linguistic, philosophical and mathematical questions, which I addressed in my dissertation.