Relations as transformations: implications for analogical reasoning

We present two experiments assessing whether the size of a transformation instantiating a relation between two states of the world (e.g., shrinks) is a performance factor affecting analogical reasoning. The first experiment finds evidence of transformation size as a significant factor in adolescent analogical problem solving while the second experiment finds a similar effect on adult analogical reasoning using a markedly different analogical completion paradigm. The results are interpreted as providing evidence for the more general framework that cognitive representations of relations are best understood as mental transformations. The question of what constitutes a relation is fundamental to much of cognition, especially for complex cognitive processes such as reasoning and problem solving (Holyoak, Gentner, & Kokinov, 2001). A case in point is analogical reasoning, which in large part involves judging the similarity between relations and structures of relations (Gentner, 1983). It follows that how relations are represented bears directly on theoretical accounts of analogy. The dominant theoretical approach to date is to treat relations as highly structured representations such as predicates with multiple arguments (e.g., Gentner, 1983), or as combinations of objects bound to actor or patient roles (e.g., Hummel & Holyoak, 1997). In contrast, several authors now argue that mental representation of relations can best be understood as representations of a transformation between two states of the world (but see Larkey & Markman, 2005, for a critical discussion of this approach). For example, Hahn, Chater, and Richardson (2003) have proposed a metric of similarity based on the number of steps it takes to transform one entity into another. Two entities are viewed as highly similar if a single operation can transform one into the other and increasingly less dissimilar the more operations are needed. One consequence of Hahn et al.'s approach is that relational similarity, and so analogical reasoning, is conceived of in terms of transformations. In a similar vein, Thomas and Mareschal (1997) demonstrated how viewing similarity as transformation provides a parsimonious explanation of asymmetries in similarity judgements. Thomas and Mareschal (2001) applied their similarity as transformation approach to metaphor interpretation (closely related to analogical reasoning). In addition to general accounts of similarity as transformation, there are also extant, more specific, mechanistic accounts of cognitive processes where relations are represented as transformation. For instance, Rogers and McClelland (2004) present a version of Rumelhart and Todd's (1993) connectionist model of semantic cognition wherein relations (e.g., IS, HAS) modulate the mappings between objects and attributes (e.g., bird HAS feathers). In such an account, a relation is actually instantiated as a transformation from one semantic entity (e.g., bird) to another (e.g., feathers). The current work focuses on the size of transformations involved in simple analogies. While there has been substantial work showing that the surface similarity between two domains affects the likelihood that an analogy will be drawn (e.g., Gick & Holyoak, 1983; Novick, 1988; but see Blanchette & Dunbar, 2000; 2001, for other factors such as audience characteristics and goals on analogical reasoning as well as the important distinction between analogical retrieval and generation), there is little work on the effects of the relational similarity between two domains. One reason is that in the classical view of relations it is difficult to quantify the similarity between two different relations. However, if relations are viewed as mental representations of transformations, similar relations will have similar transformational effects. Thus two relations that change the state of the world in similar ways will be more similar than two transformations that change the state of the world in different ways. With regard to analogical completion, we propose that the size of a transformation should be one determinant of successful analogical reasoning involving that transformation. The central idea is that when there is a large enough transformation (i.e., relation), there is less overlap between the representations of the objects instantiating the transformation, thereby making that transformation less confusable with others—and consequent analogical reasoning more successful. In the following two experiments we investigate whether transformation size is a performance factor in analogical reasoning, as might be expected if relations are transformations. The results of these experiments are especially informative because other major accounts of analogical reasoning and development (e.g., Gentner, 1983; Hummel & Holyoak, 1997; Hofstadter & the Fluid Analogies Research Group, 1995) do not conceive of relations as transformations and make no obvious predictions about the impact of transformation size. In what follows, transformation size is defined as the distance between two concepts within semantic similarity space. That is, a transformation is considered to be large when it involves two objects or states of objects that are very dissimilar and small when it involves object states that are similar. Experiment 1 establishes that transformation size is a factor in predicting the likelihood of analogical completion. Here, we employ Gick and Holyoak's (1983) renowned analogical problem-solving procedure. These experiments use a version of Dunker's radiation problem where participants are asked how best to use radiation to destroy a life-threatening tumour without damaging the patient. The desired answer involves using many low-intensity rays converging on the tumour from different directions. Without prior knowledge or help high-school students were reported to provide the correct solution only 10% of the time. However, when participants were first given a story involving a different problem but with a structurally similar solution, the percentage of correct solutions to the radiation problem increased to approximately 30%. Transformation size is manipulated by modulating the size of the effect of the radiation from destroying the tumour to shrinking the tumour. Experiment 2 replicates these results using a different form of analogy (the more classic item analogy) with multiple exemplars. The second experiment also includes measures of the size of semantic differences between items of a relation and rules out additional factors such as word frequency