STRUCTURAL ALIGNMENT IN COMPARISON: No Difference Without Similarity

Theories of similarity generally agree that the similarity ofa pair increases with its commonalities and decreases with its differences. Recent research suggests that this comparison process involves an alignment of structured representations yielding commonalities, differences related to the commonalities, and differences unrelated to the commonalities. One counterintuitive prediction of this view is that it should be easier to find the differences between pairs ofsimilar items than to find the differences between pairs of dissimilar items. This prediction is particularly strong for differences that are related to the commonalities. We tested this prediction in two experiments in which subjects listed a single difference for each of a number of word pairs. The results are consistent with the predictions of structural alignment. In light of these findings, we discuss the potential role of structural alignment in other cognitive processes that involve comparisons. The process of determining the similarity of a pair of items is central to diverse mental processes, including categorization (Smith & Medin, 1981), problem solving (Novick, 1990; Ross, 1987), and affect (Kahneman & Miller, 1986). The general consensus of research on similarity is that a pair's similarity increases with its commonalities and decreases with its differences (Tversky, 1977). Much recent research extends this general point by suggesting that similarity comparisons involve a process of structural alignment akin to the comparison process involved in analogy (Gentner & Markman, in press; Goldstone, Medin, & Gentner, 1991; Markman & Gentner, 1993a, 1993b; Medin, Goldstone, & Gentner, 1993). This view assumes that mental representations consist of hierarchical systems that encode objects, attributes of objects, relations between objects, and relations between relations. These structured representations may encode information about perceptual as well as conceptual relations. For example, the configurations in Figure la could be represented by the structural representations depicted in Figure lb. As in the structure-mapping theory of analogy (Gentner, 1983), we assume that the process of comparison is one of structural alignment between two mental representations to find the maximal structurally consistent match between them. A structurally consistent match is one that satisfies the constraints of parallel connectivity and one-to-one mapping (Falkenhainer, Forbus, & Gentner, 1989; Gentner, 1983, 1989; Holyoak & Thagard, 1989; Keane, 1988; Markman & Gentner, 1993b; Medin, Goldstone, & Gentner, 1990, 1993). Parallel connectivity Address correspondence to Dedre Gentner, Northwestern University, Department of Psychology, 2029 Sheridan Rd., Evanston, IL 60208. 152 Copyright ® 1994 American Psychological Society says that if two predicates are matched, then their arguments must also match. For example, if the "above" relations in the configurations in Figure 1 are matched, then the two items on top must be placed in correspondence (the striped circle with the striped square), and likewise the two items on the bottom must be placed in correspondence (the checked square with the checked circle). One-to-one mapping requires that each element in one representation correspond to at most one element in the other representation. Thus, in Figure 1, the circle in the lefthand configuration could not be placed in correspondence simultaneously with both the circle and the square in the righthand configuration. As this example illustrates, in many cases more than one structurally consistent interpretation is possible for a given comparison. Here, on one interpretation, the commonality is that both configurations contain circles. On another interpretation, the commonality is that both have something above something else. According to the systematicity principle (Gentner, 1983, 1989), when there are multiple interpretations of a pair, all else being equal, the one that preserves the maximal (i.e., largest and deepest) connected relational structure is preferred (Forbus & Gentner, 1989; Gentner & Landers, 1985; Gentner, Rattermann, & Forbus, 1993). This interpretation can then be used to calculate a similarity rating for the items or to subserve some other cognitive process that requires a comparison. On this account, the commonalities and differences of a pair are determined relative to an interpretation (Falkenhainer et al., 1989; Markman & Gentner, 1993a; Medin et al., 1993). The commonalities are simply the elements of the matching representational structure. For example, the best interpretation of the comparison in Figure 1 might involve placing the "above" relations in correspondence, making that relation a commonality. The differences are separated into two types: those related to the common structure (called alignable differences) and those not related to the common structure (called nonalignable differences). For example, on this interpretation, the fact that the circle is on top in one configuration while the square is on top in the other is an alignable difference, because the circle and square are nonidentical elements placed in correspondence by virtue of their like roles in matching structures. In contrast, the triangle in the left-hand configuration is a nonalignable difference, because it does not correspond to anything in the righthand configuration. To put it another way, alignable differences arise from and are connected to the common structure, whereas nonalignable differences are independent of the common structure. This proposal is related to previous suggestions that similarity focuses primarily on the commonalities of a pair (Krumhansl, 1978; Sjoberg, 1972). Our proposal goes beyond this idea in suggesting that not only are commonalities central, but even VOL. 5, NO. 3, MAY 1994 the differences that are considered are those related to the commonalities (i.e., the alignable differences). In other studies, we provided evidence for this view by asking subjects to list the commonalities and differences of word pairs (Markman & Gentner, 1993a). We found that subjects could list more commonalities for similar pairs of words than for dissimilar pairs. However, the reverse did not hold: Subjects did not list more differences for dissimilar pairs than for similar pairs. Instead, as we would expect if commonalities and alignable differences are deeply related, subjects listed more alignable differences for similar pairs than for dissimilar pairs and more nonalignable differences for dissimilar pairs than for similar pairs. The total number of differences was roughly consistent across similarity. Further sorting tasks revealed conceptual relationships between the commonalities and the alignable differences, but not between the commonalities and the nonalignable differences, bearing out the claim that alignable differences are related to commonalities but nonalignable differences are not. Indeed, the number of commonalities was positively correlated with the number of alignable differences, but not with the number of nonalignable differences. Finally, there was evidence that alignable differences are considered more important than nonalignable differences in that subjects listed more alignable differences than nonalignable differences overall. The idea that alignable differences are more salient in the comparison process than are nonalignable differences has one startling implication. Because there are more commonalities for similar pairs than for dissimilar pairs, there should also be more alignable differences for similar pairs than for dissimilar pairs. Thus, if, as suggested by the previous findings, subjects find it Fig. 1. Illustration of the role of alignment in comparison. The geometric configurations (a) can be encoded by the structured representations in (b). In these structured representations, ovals denote relations, rounded boxes denote objects, and bold rounded boxes denote attributes. "Med" denotes "medium sized." VOL. 5, NO. 3, MAY 1994 PSYCHOLOGICAL SCIENCE Dedre Gentner and Arthur B. Markman Fig. 2. Venn diagrams illustrating the comparison of a similar pair and a dissimilar pair. easier to report commonalities for high-similarity pairs than for low-similarity pairs, then they should generally be able to find differences (at least alignable differences) more easily for similar pairs than for dissimilar pairs. The idea that differences are easier to find for similar pairs than for dissimilar pairs runs against the plausible intuition that differences should be easier to find the more different the pair. We schematize this intuition in Figure 2, borrowed from Tversky (1977). The objects' representations are the sets of properties represented by the circles, and the match between representations corresponds to the overlap of the sets. This diagram suggests that all objects are represented in equal detail and that there should be more differences for dissimilar pairs than for similar pairs provided that the commonalities and differences are independent of one another. Thus, differences should be easier to find for dissimilar pairs than for similar pairs.' The two studies we present here examine the claim that subjects should find differences more easily for pairs of similar items than for pairs of dissimilar items. In the first experiment, subjects saw 40 word pairs on a sheet of paper and were told to write one difference for as many different pairs as they could. They were told that they would not have enough time to respond to all of the pairs, so they should try to do the easy pairs first. Half the pairs were of high similarity and half were of low similarity (based on intuitions that were confirmed by subjects' similarity ratings). The predictions are straightforward, though counterintuitive. If similarity comparisons involve structural alignment, then subjects should list more differences (and particularly more alignable differences) for similar pairs than for dissimilar pairs.