Tversky Contrast Model

[Tversky's (1977) contrast model](/en/Tversky's%20(1977)%20contrast%20model) is a theoretical framework that shows that similarity judgments made by humans do not necessarily follow "triangular inequalities," unlike typical distance measures such as Euclidean distance. Triangular inequalities are properties of distance and similarity that state that "the distance (or dissimilarity) between A and C is not greater than the sum of the distances between A and B, and between B and C" when considering distance and similarity. However, Tversky showed, both experimentally and theoretically, that this rule does not apply to human similarity judgments.

In the contrast model, similarity is expressed by subtracting the effect of features specific to A (A-B) and B (B-A) from the amount of shared features (A∩B) between objects A and B. Here, features are not counted with uniform weights, but selective and asymmetric weighting occurs depending on the context and the observer's perspective. This asymmetry and context-dependence is one reason why similarity deviates from a "symmetric and distance measure".

In Tversky & Gati (1982), experiments were reported that directly demonstrate the violation of such triangular inequalities. For example, they prepared three concepts A, B, and C within a category and obtained results such as "A and B are similar" and "B and C are also similar," even though they knew that "A and B are similar" and "B and C are not equally similar. This suggests that human judgments of similarity are strongly influenced by comparisons between features and contextual selection, rather than by simple "distance" thinking. Therefore, there are aspects of similarity judgments that cannot be explained by the geometric distance model (a model that uses the distance between points in dimensional space), and this non-Euclidean mode of judgment is considered to be one of the essential features in human conceptual recognition and categorization.

Isn't it different from "non-Euclidean" to say that distance is not valid?

Yes, it is. "Distance is not valid (distance property, especially triangle inequality, is not valid)" and "non-Euclidean (non-Euclidean geometry)" are two different concepts. - Distance is not valid: It means that the similarity judgment does not satisfy the mathematical concept of "distance". If it does not satisfy the basic conditions as a distance (non-negativity, identity, symmetry, triangular inequality), it can no longer be called a "distance" and loses its structure as a "distance space". - Non-Euclidean: Non-Euclidean geometry refers to spatial structures for which the axioms of Euclidean geometry (especially the fifth canon on parallel lines) do not hold. Even in non-Euclidean geometry, there are usually measures that can be defined as "distances" and trigonometric inequalities hold (the appropriate distance function still satisfies the axiom as "distance").

In other words, the result shown by Tversky et al. that "similarity violates the triangular inequality (i.e., breaks its establishment as a distance axiom)" does not mean that the result of the similarity judgment shows a "non-Euclidean distance," but that it "cannot be formulated as a distance to begin with. Since a non-Euclidean distance is only a "distance," the triangular inequality is protected, but Tversky's argument is problematic because of its "not even distance" point.

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