How many possible languages are there?

We adopt a description of natural languages in terms of strings of crosslinguistically variable syntactic features (parameters), complying with a spec- ified hypothesis of "universal grammar", and we deal with two problems: first, as- sessing the statistical significance of language distances calculated on the basis of such features and recently used to reconstruct phylogenetic trees; second, counting the minimal overall number of possible human languages, i.e. of strings satisfy- ing the implicational rules which describe dependencies between parameters of the specified universal grammar. In order to accomplish these tasks, we had to develop a sampling algorithm capable of dealing correctly with such rules. The potential significance of these results for historical and theoretical linguistics is then briefly highlighted.