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Psychological Review J Copyright © 1977 C ? by the American Psychological Association, Inc Psychological Review J Copyright © 1977 C_? by the American Psychological Association, Inc. VOLUME 84 NUMBER 4 JULY 1977 Features of Similarity Amos Tversky Hebrew University Jerusalem, Israel The metric and dimensional assumptions that underlie the geometric represen- tation of similarity are questioned on both theoretical and empirical grounds. A new set-theoretical approach to similarity is developed in which objects are represented as collections of features, and similarity is described as a feature- matching process. Specifically, a set of qualitative assumptions is shown to imply the contrast model, which expresses the similarity between objects as a linear combination of the measures of their common and distinctive features. Several predictions of the contrast model are tested in studies of similarity with both semantic and perceptual stimuli. The model is used to uncover, analyze, and explain a variety of empirical phenomena such as the role of common and distinctive features, the relations between judgments of similarity and differ- ence, the presence of asymmetric similarities, and the effects of context on judgments of similarity. The contrast model generalizes standard representa- tions of similarity data in terms of clusters and trees. It is also used to analyze the relations of prototypicality and family resemblance. Similarity plays a fundamental role in errors of substitution, and correlation between theories of knowledge and behavior. It serves occurrences. Analyses of these data attempt to as an organizing principle by which individuals explain the observed similarity relations and classify objects, form concepts, and make gen- to capture the underlying structure of the ob- eralizations. Indeed, the concept of similarity jects under study. is ubiquitous in psychological theory. It under- The theoretical analysis of similarity rela- lies the accounts of stimulus and response tions has been dominated by geometric generalization in learning, it is employed to models. These models represent objects as explain errors in memory and pattern recogni- points in some coordinate space such that the tion, and it is central to the analysis of con- observed dissimilarities between objects cor- notative meaning. respond to the metric distances between the Similarity or dissimilarity data appear in respective points. Practically all analyses of different forms: ratings of pairs, sorting of proximity data have been metric in nature, objects, communality between associations, although some (e.g., hierarchical clustering) yield tree-like structures rather than dimen- This paper benefited from fruitful discussions with sionallv. organized spaces. However, most Y. Cohen, I. Gati, D. Kahneman, L. Sjeberg, and theoretical and empirical analyses of similarity S. Sattath. assume that objects can be adequately repre- Requests for reprints should be sent to Amos Tversky, , , ... ,. , i Department of Psychology, Hebrew University, sented as points m some coordinate space and Jerusalem, Israel. that dissimilarity behaves like a metric dis- 327 328 AMOS TVERSKY tance function. Both dimensional and metric and argues that similarity should not be assumptions are open to question. treated as a symmetric relation. It has been argued by many authors that Similarity judgments can be regarded as dimensional representations are appropriate extensions of similarity statements, that is, for certain stimuli (e.g., colors, tones) but not statements of the form "a is like b." Such a for others. It seems more appropriate to repre- statement is directional; it has a subject, a, sent faces, countries, or personalities in terms and a referent, b, and it is not equivalent in of many qualitative features than in terms of general to the converse similarity statement a few quantitative dimensions. The assessment "b is like a." In fact, the choice of subject of similarity between such stimuli, therefore, and referent depends, at least in part, on the may be better described as a comparison of relative salience of the objects. We tend to features rather than as the computation of select the more salient stimulus, or the proto- metric distance between points. type, as a referent, and the less salient stimu- A metric distance function, 6, is a scale that lus, or the variant, as a subject. We say "the assigns to every pair of points a nonnegative portrait resembles the person" rather than number, called their distance, in accord with "the person resembles the portrait." We say the following three axioms: "the son resembles the father" rather than "the father resembles the son." We say "an Minimality: ellipse is like a circle," not "a circle is like an 8(a,b) > S(a,a) = 0. ellipse," and we say "North Korea is like Red Symmetry: China" rather than "Red China is like North S(a,b) = 5(b,a). Korea." As will be demonstrated later, this asym- The triangle inequality: metry in the choice of similarity statements is «(a,b) + S(b,c) > 8(a,c). associated with asymmetry in judgments of similarity. Thus, the judged similarity of To evaluate the adequacy of the geometric North Korea to Red China exceeds the judged approach, let us examine the validity of the similarity of Red China to North Korea. Like- metric axioms when 8 is regarded as a measure wise, an ellipse is more similar to a circle than of dissimilarity. The minimality axiom implies a circle is to an ellipse. Apparently, the direc- that the similarity between an object and itself tion of asymmetry is determined by the rela- is the same for all objects. This assumption, tive salience of the stimuli; the variant is however, does not hold for some similarity more similar to the prototype than vice versa. measures. For example, the probability of The directionality and asymmetry of simi- judging two identical stimuli as "same" rather larity relations are particularly noticeable in than "different" is not constant for all stimuli. similies and metaphors. We say "Turks fight Moreover, in recognition experiments the off- like tigers" and not "tigers fight like Turks." diagonal entries often exceed the diagonal Since the tiger is renowned for its fighting entries; that is, an object is identified as an- spirit, it is used as the referent rather than other object more frequently than it is identi- the subject of the simile. The poet writes "my fied as itself. If identification probability is love is as deep as the ocean," not "the ocean interpreted as a measure of similarity, then is as deep as my love," because the ocean these observations violate minimality and are, epitomizes depth. Sometimes both directions therefore, incompatible with the distance are used but they carry different meanings. model. "A man is like a tree" implies that man has Similarity has been viewed by both philoso- roots; "a tree is like a man" implies that the phers and psychologists as a prime example of a symmetric relation. Indeed, the assumption tree has a life history. "Life is like a play" of symmetry underlies essentially all theo- says that people play roles. "A play is like retical treatments of similarity. Contrary to life" says that a play can capture the essential this tradition, the present paper provides elements of human life. The relations between empirical evidence for asymmetric similarities the interpretation of metaphors and the as- FEATURES OF SIMILARITY 329 sessment of similarity are briefly discussed in ity is somewhat problematic, symmetry is ap- the final section. parently false, and the triangle inequality is The triangle inequality differs from minimal- hardly compelling. ity and symmetry in that it cannot be formu- The next section develops an alternative lated in ordinal terms. It asserts that one theoretical approach to similarity, based on distance must be smaller than the sum of two feature matching, which is neither dimensional others, and hence it cannot be readily refuted nor metric in nature. In subsequent sections with ordinal or even interval data. However, this approach is used to uncover, analyze, and the triangle inequality implies that if a is quite explain several empirical phenomena, such as similar to b, and b is quite similar to c, then the role of common and distinctive features, a and c cannot be very dissimilar from each the relations between judgments of similarity other. Thus, it sets a lower limit to the simi- and difference, the presence of asymmetric larity between a and c in terms of the similari- similarities, and the effects of context on simi- ties between a and b and between b and c. larity. Extensions and implications of the The following example (based on William present development are discussed in the final James) casts some doubts on the psychological section. validity of this assumption. Consider the simi- larity between countries: Jamaica is similar Feature Matching to Cuba (because of geographical proximity); Cuba is similar to Russia (because of their Let A = {a,b,c,...} be the domain of objects political affinity); but Jamaica and Russia are (or stimuli) under study. Assume that each not similar at all. object in A is represented by a set of features This example shows that similarity, as one or attributes, and let A,B,C denote the sets of might expect, is not transitive. In addition, it features associated with the objects a,b,c, re- suggests that the perceived distance of Jamaica spectively. The features may correspond to to Russia exceeds the perceived distance of components such as eyes or mouth; they may Jamaica to Cuba, plus that of Cuba to Russia represent concrete properties such as size or —contrary to the triangle inequality. Although color; and they may reflect abstract attributes such examples do not necessarily refute the such as quality or complexity. The character- triangle inequality, they indicate that it should ization of stimuli as feature sets has been not be accepted as a cornerstone of similarity employed in the analysis of many cognitive models.
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