Measuring Linguistic Diversity: Limits and Extensions of the Greenberg Index

Abstract

This theoretical paper proposes a refined framework for measuring linguistic diversity in multilingual societies, with a focus on the Greenberg Diversity Index (GDI), a widely used indicator based on the probability that two randomly selected individuals speak different languages. After deriving exact upper and lower bounds for GDI under varying language distributions, the paper introduces several theoretical extensions that capture more realistic sociolinguistic configurations. These include competence-weighted indices to account for individual multilingualism, co-competence matrices to model overlapping language repertoires, distance-weighted formulations based on interlinguistic similarity, and entropy-based metrics that reflect the unpredictability of language identity. Each index is analysed in terms of its mathematical structure, interpretability, and alignment with different types of linguistic data. A comparative framework is proposed to support the selection and interpretation of these metrics across research and policy applications. Together, these contributions advance the quantitative modelling of linguistic diversity and offer tools for more nuanced sociolinguistic analysis.

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