What counts? Visual and verbal cues interact to influence what is considered a countable thing.

Many famous paintings illustrate variations in what we here dub "referential depth." For example, paintings often include not only portrayals of uniquely referenced items, but also reflections of those items in mirrors or other polished surfaces. If a painting includes both a dancer and that dancer's reflection in a mirror, are there one or two dancers in the painting? Although there are two images of a dancer, both images reference the exact same dancer. Consequently, counting both may seem to violate the constraint against double counting (Gelman & Gallistel, 1978). This illustrates that determining which things "count" in a given context may not be straightforward. Here we used counting tasks paired with illustrations that manipulated referential depth to investigate the conceptual, perceptual, and language variables that may influence whether a "thing" is a "countable thing." Across four experiments, 316 participants counted items in displays that included both foreground items and items placed inside mirrors, picture frames, and windows. Referential depth and frame boundaries both influenced counting: For one thing, participants were more likely to count items contained by windows than by picture frames or mirrors. Moreover, items in mirrors were rarely counted unless they were interpreted as reflections of items "off screen." Also, the items contained inside windows were sometimes (~10% of trials) excluded from the counts, when counting them would require crossing frame boundaries. We concluded that conceptual and perceptual contexts both influence people's decisions about the physical boundaries of the to-be-counted set and which items within these boundaries are countable.