What Can Be Learned from Vagueness

Speakers disagree about where the boundaries of graded words such as "tall" lie, and the resulting vagueness is usually counted as a cost to the listener. We ask what the same disagreement offers a learner that reads only text and never sees the magnitudes the words describe. When speakers draw a boundary at slightly different points, objects near the boundary receive the names of both neighboring categories. In a simple model, averaged over all namings, these misnamings add extra squared error for the listener that grows with the square of the boundary spread, while the fraction of adjacencies an idealized learner recognizes grows with the spread itself (Proposition 1). We test four predictions with language models trained only on naming sentences. In an artificial language with a known scale, the learner's internal representations encode no order among the categories when all speakers share one set of boundaries, and encode a geometry that matches the scale once the boundaries differ slightly. In the color naming of 92 languages of the World Color Survey, redirecting departures from the majority term at random, while keeping where and how often speakers depart, weakens the color geometry recovered from naming in every language, and majority-vote aggregation erases the geometry. English probability and quantity words, larger transformers, an LSTM and a pretrained model give results in the same direction. Variation in naming thus supplies relations between categories that no single sentence states.