Infinitely productive language can arise from chance under communicative pressure

Human communication is unparalleled in the animal kingdom. The key distinctive feature of our language is productivity: we are able to express an infinite number of ideas using a limited set of words. Traditionally, it has been argued or assumed that productivity emerged as a consequence of very specific, innate grammatical systems. Here we formally develop an alternative hypothesis: productivity may have rather solely arisen as a consequence of increasing the number of signals (e.g. sentences) in a communication system, under the additional assumption that the processing mechanisms are algorithmically unconstrained. Using tools from algorithmic information theory, we examine the consequences of two intuitive constraints on the probability that a language will be infinitely productive. We prove that under maximum entropy assumptions, increasing the complexity of a language will not strongly pressure it to be finite or infinite. In contrast, increasing the number of signals in a language increases the probability of languages that have—in fact—infinite cardinality. Thus, across evolutionary time, the productivity of human language could have arisen solely from algorithmic randomness combined with a communicative pressure for a large number of signals.