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

ABSTRACT

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.

KEYWORDS:

• Concepts
• Language
• Categorization
• Prototypes
• Inference
• Language and thought

Notes

¹The following exchange occurred at the closing of a round-table debate on the subject of “Fodor's Puzzle” of Concept Learning at the 2005 Meeting of the Cognitive Science Society:

Andy Clark: I think Jerry Fodor is getting away with something by slipping it in very early into the argument. And what comes in very early is this rejection of pragmatism. The whole argument is just going to go through flawlessly and beautifully, if you agree that there is no essential connection between grasping a concept and being able to do things in the world. There's all sorts of ways to act in the world and learning to do things in the world, which are completely immune to the worries that Jerry Fodor is raising. So that the only way that this argument goes through is if you buy something that is very counterintuitive, that is to say, that there is no essential relation between the concepts that you grasp, and the things that you can actually do in the world. And that's the price that Jerry has to pay for the very lovely argument.

Jerry Fodor: There's no price for me. I'm a Cartesian!

The full transcript can be found at the link below, courtesy of Jesse Snedeker. http://www.wjh.harvard.edu/~lds/pdfs/Niyogi_Snedeker-2005.pdf

²I will sometimes refer to this state as a prototype. On the present formulation, a prototype is not a “thing” but rather a distributed neural activation pattern that (by definition) overlaps more with patterns elicited by more typical than with those elicited by less typical exemplars (see General Discussion).

³Of course our participants know that three-sided polygons are triangles and, as discussed below, tend to interpret phrases like “three-sided figure” as denoting a triangle. But reflective knowledge and the ability to strategically translate from the first phrase to the second does not mean that the two will elicit the same representational state, just as knowing that dogs bark does not mean that a bark and the word “dog” activate the same representational state (Lupyan & Thompson-Schill, 2012). That all participants know and use the term “triangle” make the experiments that follow an especially strong test of the stated hypotheses because any observed differences exist despite participants’ ability to translate between these functionally equivalent expressions. If these putatively coextensive cues activate detectibly different representations, then one may infer the differences to be even larger in the case when a suitable category name is available and in cases where a name is not available.

⁴Indeed, Elio Motors faces a fascinating legislative situation: Their vehicle is federally classified as a motorcycle because it does not have four wheels, allowing it to meet more lenient emission standards set for motorcycles. At the same time, the company has been (successfully) lobbying to obtain an exemption on motorcycle licence or helmet use requirements for the drivers of the vehicle on the basis that it is really a car.

⁵The data were aggregated prior to analysis to have a single RT mean for each Subject × triangle-type (equilateral/isosceles/scalene) × Orientation (canonical vs. non-canonical) to ensure that each trial type was weighed equally by the regression.

⁶The resulting representation may be called a prototype: a distributed pattern of neural activity that is more overlapping with some depictions of triangles (the typical/canonical ones) than others (the atypical/non-canonical ones).

⁷Experiments on prototype effects have sometimes been seen to be arguing that a learned category is represented as the prototype (Osherson & Smith, 1981; cf. Laurence & Margolis, 1999). Posner and Keele's (1968) actual findings showed that although people showed sensitivity to the unobserved visual template, it was judged to be less familiar than actually observed exemplars—a result clearly at odds with the view that the template is the learned representation of the category. The idea that embracing prototype effects implies committing to a theory in which the prototype serves as the representation of the category reflects a misunderstanding. In Rosch's own words:

“To speak of a prototype at all is simply a convenient grammatical fiction; what is really referred to are judgments of degree of prototypicality. Only in some artificial categories is there by definition a literal single prototype. . . . For natural-language categories, to speak of a single entity that is the prototype is either a gross misunderstanding of the empirical data or a covert theory of mental representation (Rosch, 1978, p. 40).

⁸Much of this work involved contrasting “prototype” and “exemplar” models (see Mack, Preston, & Love, 2013, for recent formulation). Although often presented in opposition, the strengths of both formalisms are neatly incorporated into connectionist/connectionist-inspired architectures (Kruschke, 1992; Love, Medin, & Gureckis, 2004; Rogers & McClelland, 2004) in which a prototype is an emergent rather than a fixed entity. It may be tempting to interpret the present results in terms of people accessing different exemplars in different contexts—for example, accessing an exemplar of “equilateral triangle” when hearing or reading “triangle”. But it is difficult to see how invoking exemplar theory in this way offers a genuine alternative explanation to what is proposed here.

⁹This under-representation of right triangles may well disappear if people are asked to draw “maths triangles” or are even primed with a mathematical context, a finding that would further corroborate the flexible nature of formally defined categories.

¹⁰In a recent presentation, Greene (2014) discussed a task in which participants were asked to indicate what objects never occur as part of certain scene categories. For example, “What do you never find in a kitchen?” Such questions would appear to be completely under-determined as there is an infinite number of objects one does not find in a kitchen. Yet for many categories, participants had a shockingly high amount of agreement (e.g., no toilets in a kitchen, no skyscrapers in a field). On the view advocated here, these responses are indicative of implicit contrast categories in action: If a common contrast category of kitchen is “bathroom”, and “toilet” is a common object in the latter, then it is a salient non-member of the former.

¹¹A typical adult faced with this procedure might quickly conclude that what is being demanded is the category, rather than any specific instance. It is my contention that a rapid realization of this sort is mediated—implicitly or explicitly—by verbal labels. Absent the labels, the participant may be guided much more by overall similarity to provided exemplars (Perry & Lupyan, 2014).

¹²Similar observations were made much earlier by Tversky and Kahneman in their work on the availability heuristic (e.g., Tversky and Kahneman, 1974), and it is likely that there is considerable overlap in the mechanisms underlying both types of effects.