Recursion, Infinity and Modeling

Hauser (et al.) [1] claim that a core property of the human language faculty is recursion and that this property “yields discrete infinity” of natural languages. On the other hand, recursion is often motivated by the observation that there are infinitely many sentences that should be generated by a finite number of rules. It should be obvious that one cannot pursue both arguments simultaneously, on pain of circularity. We want to argue that discrete infinity is not derived, but a modeling choice. Thus, discrete infinity is not an empirical fact about natural languages (see [2, 3] for arguments related to that point). Attributing discrete infinity to natural languages confuses a model with empirical reality. A common motivation for assuming an infinite model is that assuming a finite model implies that there is a least upper bound on sentence length. Although there may be performance considerations to believe that there is an upper bound, there seem to be no grammatical reasons for assuming a least upper bound. It is certainly true that recursive rules yield an infinite model (infinitely many sentences). On the other hand, if we want to generate an infinite set with a finite number of rules, these rules need to include some form of recursion. Since there are infinite sets that cannot be generated by a finite set of rules, the interesting direction is not to derive discrete infinity from recursion, but to assume infinity with a finite description to yield recursion. Even if natural languages actually be finite (whatever that may mean), there are reasons to believe that the human language faculty contains recursion. As was pointed out by Savitch [4], there are finite (formal) languages which can be more economically described as subsets of an infinite language, i.e. using recursion. Strings that are “too long to be grammatical” (again, whatever that means) would essentially be the result of the recursive rules overgenerating.