A Mathematical Model of the Loss of Verb-Second in Middle English

Lightfoot (1999) proposes the following explanation for the loss of the verb-second rule in Middle English: There were two regional dialects of Middle English, a northern dialect influenced by Old Norse with a verbsecond rule, and a southern dialect with a slightly different word order. Children acquire the verb-second rule based on hearing some critical fraction of cue sentences requiring such a rule. As the dialects experienced increased contact, northern children were less likely to hear enough cue sentences, and consequently acquired a different grammar, resulting in the extinction of the northern dialect. This hypothesis can be modeled with differential equations. By using dynamical systems methods, the catastrophe in question may be modeled by a mathematical event known as a saddle-node bifurcation. A key part of the model is the function that gives the probability of learning the northern dialect given that a fraction of the local population uses it. Other model acquisition algorithms, such as memoryless learner (Niyogi & Berwick 1996), give the mysterious result that verb-second languages should be extremely stable, in contrast to the history of English. This new model provides an explanation for that behavior: Memoryless learners are more sensitive to noise, resulting in a differently shaped function that does not allow the northern grammar to disappear. This model q ) (x q x demonstrates how dynamical systems theory can be used to study language change and learning models.