The paradox of the universal triangle: Concepts, language, and prototypes

For over 300 years, the humble triangle has served as the paradigmatic example of the problem of abstraction. How can we have the idea of a general triangle even though every experience with triangles is with specific ones? Classical cognitive science seemed to provide an answer in symbolic representation. With its easily enumerated necessary and sufficient conditions, the triangle would appear to be an ideal candidate for being represented in a symbolic form. I show that it is not. Across a variety of tasks-drawing, speeded recognition, unspeeded visual judgments, and inference-representations of triangles appear to be graded and context dependent. I show that using the category name "triangle" activates a more prototypical representation than using an arguably coextensive cue, "three-sided polygon". For example, when asked to draw "triangles" people draw more typical triangles than when asked to draw "three-sided polygons". Altogether, the results support the view that (even formal) concepts have a graded and flexible structure, which takes on a more prototypical and stable form when activated by category labels.