Abstract
The Constant Rate Hypothesis (Kroch 1989) states that when grammar competition leads to language change, the rate of replacement is the same in all contexts affected by the change (the Constant Rate Effect, or CRE). Despite nearly three decades of empirical work into this hypothesis, the theoretical foundations of the CRE remain problematic: it can be shown that the standard way of operationalizing the CRE via sets of independent logistic curves is neither sufficient nor necessary for assuming that a single change has occurred. To address this problem, we introduce a mathematical model of the CRE by augmenting Yang’s (2000) variational learner with production biases over an arbitrary number of linguistic contexts. We show that this model naturally gives rise to the CRE and prove that under our model the time separation possible between any two reflexes of a single underlying change necessarily has a finite upper bound, inversely proportional to the rate of the underlying change. Testing the predictions of this time separation theorem against three case studies, we find that our model gives fits which are no worse than regressions conducted using the standard operationalization of CREs. However, unlike the standard operationalization, our more constrained model can correctly differentiate between actual CREs and pseudo-CREs—patterns in usage data which are superficially connected by similar rates of change yet clearly not unified by a single underlying cause. More generally, we probe the effects of introducing context-specific production biases by conducting a full bifurcation analysis of the proposed model. In particular, this analysis implies that a difference in the weak generative capacity of two competing grammars is neither a sufficient nor a necessary condition of language change when contextual effects are present.
Highlights
In dynamical-systems terminology, the production bias mechanism induces a bifurcation in the parameter space of the extended model: small tweaks made to either the biases or to the proportion of input falling in each context can alter the trajectory of language change entirely by determining which of the two grammars will win out (Fig. 8)
Hawkins’s (2004:38) principle of Minimize Domains states that “[t]he human processor prefers to minimise the connected sequences of linguistic forms and their conventionally associated syntactic and semantic properties in which relations of combination and/or dependency are processed.”. This general principle is made concrete using a metric of Early Immediate Constituents (EIC), which serves to favour syntactic structures with a uniform directionality of branching
Building on earlier work that derives logistic evolution as a population-level property of language change (Niyogi and Berwick 1997; Yang 2000, 2002), we have provided a mechanism for the Constant Rate Effect proposed by Kroch (1989)
Summary
We propose to augment Yang’s (2000) model of grammar competition with a set of production biases bi which modulate the grammar weights pt and qt in actual linguistic production. The best one can do is to iterate the model for various choices of model parameter values and initial conditions and compare the resulting trajectories against empirical data, an approach which soon becomes computationally prohibitive as the number of logically possible model parameter combinations grows as a superlinear function of the number of model parameters To tackle this problem, we will conduct a full analysis of the behaviour of our model in the limit t → ∞ and show that, under most empirically meaningful combinations of model parameter values, the underlying trajectory qt is well approximated by a logistic curve. A reader who is willing to skip the technicalities of the logistic approximation may advance straight to Sect. 4
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