To explore the phenomenon of transcategorial changes and functionalization, we use a model in which a finite population of language users exchange meta vocables1.11 with their immediate neighbours. An increase in activity in the exchange of these meta vocables determines whether a transcategorial change will occur. In our model the meta vocables are either in a lexical state or in a functionalized state.
There are many ways to simulate the statistical nature of a system, including phase transitions. Since we are in uncharted territory where the simulation of phase transition in language is concerned, we will make as few assumptions as possible in the modeling process. Statistical mechanics provides an appropriate model that is simple, well-known and mathematically tractable. It is called the Ising model[51].
The basic two-dimensional Ising model demonstrates a second-order phase transition. As we have explained, a second-order phase transition is characterized by a smooth change from one state to another. With a slight modification, (adding a degeneracy feature), it demonstrates a first-order phase transition. Using this version of the Ising model, we show that the nature of transcategorial change between a lexical state and a functional state can be described by a first-order phase transition.
The Ising model consists of a number of individual constituents, usually referred to as spins. Spins can be understood as simple vectors with discrete orientations, either up and down, but computationally can be modeled as binary states, either 0 or 1. This simplicity is what makes the Ising model so attractive as a model. A first-order phase transition is implemented by introducing two new features to the system that are mathematically simple. One is implemented macroscopically as a bias field that causes one state to be energetically favoured over the other - that is, one of the two states is lower in energy than the other. The second is implemented as a degeneracy in the spin states. This requires further explication.
Physics informally defines degeneracy as a loss of feature. We have mentioned that constituents in our model can be in one of two states. In simple terms, if each state is treated similarly (bias field aside), then each state has the same statistical likelihood of being present - in the case of two states, that is 50%. If we now ascribe three possible states for constituents rather than two and each is equally likely, then the probability is reduced to 33% for each given state. But if two of these states are effectively identical and indistinguishable as far as the model is concerned, then a bias of 66% (33% each for the two identical states) is generated in the state potential. Though the underlying statistics do not change, two of these states are interchangeable. Such identical states are described as being degenerate because of their lack of discriminating features. In the case of our language theory, the preferred (or biased) state of vocables is the lexical and the degenerate state is the functional. We will elaborate later on how vocables exhibit degeneracy.
In a physical system such as a system of H2O molecules, these biases will result in preferred states such as solid/liquid/vapor depending on the circumstances. H2O molecules will tend to a solid state when there is little energy available to the system while they will tend to a liquid state when more energy is available to it, and finally a vapour state at even higher energy levels. Each state has a structural coherence associated with it that represents an energetically preferred configuration (e.g. crystalline for the solid state) and plays the role of a biasing force. At certain system energies (i.e. temperature), the H2O system tends to a particular structure. Within the Ising model, the energy is specific to the temperature of the system and can be controlled explicitly by tuning the system relative to a critical point where the two biases balance. By then coupling the spins energetically to their neighbours via a nearest neighbour interaction, the system is thus equipped to demonstrate a first-order phase transition.
As mentioned, these descriptions rely on the assumption that the entire system is at least at quasi-equilibrium (not changing rapidly). The rate at which the constituents interact must be much shorter than the rate at which any changes take place. The time scales involved with structural changes in vocables range from a few generations in the cases of functionalization or in the emergence of creole languages, to a hundred years or more for the emergence of the natural language phenomenon. Compare this with the timescale for vocable use, something on the order of seconds or minutes. Clearly the applicability of the Ising model in this regard is strong.
Our Ising model hosts meta-vocables as constituents. The nearest neighbour interaction in our model is a simplified version of what may occur in a real population, as individuals influence each other's use of vocables within linguistic transactions. The activity (comparable to energy) variable in the system regulates the flow of exchange while the capacity to generate effects (comparable to degeneracy) regulates the individual influence of language users on each other. The activity in the system is an abstract concept and can be understood mainly as the level of exchange but also as propagation through a finite population.
As the level of activity increases in the system, it is more likely that functionalization will occur. While the nature of the activity assumes that the exchange of specific vocables is homogenous, we think that in reality some vocables will be in use more often than others. We might be tempted to assume that some vocables require less use over time to become functionalized because of what they mean or because of their initial syntactic role. But we doubt the validity of this assumption. For example, have will become functionalized while running will not despite the fact that they are both verbs. The difference lies in the frequency of use in linguistic interaction. The attenuation of run beyond running in say, running a fever, is not likely to happen because the occurrence of the particular run vocable in linguistic transactions is not as frequent as the occurrences of have. To compensate for this over simplification, we have implemented a feature, comparable in role to potential degenerate states - which we call potential attenuated states. In a standard Ising model the number of potential degenerate states is equal for all constituents. This feature illustrates the idea of attenuation in vocables. Since, in our model, we lack a certain control over the level of activity that each meta vocable may be engaged in, we have implemented various potential attenuation states. These illustrate that vocables are individually potentially more or less susceptible to functionalization.
With this experiment, we aim to demonstrate that it is the number of instances in the lifespan of a particular vocable that determines structural changes rather than its semantic or syntactic type. Moreover, we will demonstrate that the use of the simplest model that describes first-order phase transition - the Ising model - is relevent in the explanation of such a phenomenon, though somewhat limited. More sophisticated models will subsequently be explored in which phase transitions will also be considered as altering phenomena. But more importantly, if the process of functionalization is scalable to the larger phenomenon of grammaticalization, it would seem that, as an important structural change, grammaticalization can emerge spontaneously from proto-language, given a certain level of proto-language use across a critical number of individuals. Of course, the time scale of what we consider ``spontaneous'' in the evolution of language remains to be clearly established. We will also point out that these findings are consistent with Bickerton's in suggesting that there are no transitional steps in the emergence of a natural language from a proto-language.