Deriving the Wug-Shaped Curve: A Criterion for Assessing Formal Theories of Linguistic Variation

In this review, I assess a variety of constraint-based formal frameworks that can treat variable phenomena, such as well-formedness intuitions, outputs in free variation, and lexical frequency-matching. The idea behind this assessment is that data in gradient linguistics fall into natural mathematical patterns, which I call quantitative signatures . The key signatures treated here are the sigmoid curve , going from zero to one probability, and the wug-shaped curve , which combines two or more sigmoids. I argue that these signatures appear repeatedly in linguistics, and I adduce examples from phonology, syntax, semantics, sociolinguistics, phonetics, and language change. I suggest that the ability to generate these signatures is a trait that can help us choose between rival frameworks.