Meta-analogical transfer: Transfer between episodes of analogical reasoning.

Meta-analogical transfer (i.e., transfer due to forming an analogy between analogies) was demonstrated in four experiments. Results suggested that the basis of meta-analogical transfer was transfer of predicate mappings (mappings of the concepts used to represent analogies) between separate episodes of analogical reasoning. Episodes of letter-string analogy problem solving of the form, “If abc was changed into abd, how would you change kji in the same way?” were used. These analogy problems are well-constrained and have multiple plausible solutions. In Experiment 1 participants generated solutions in two separate analogical reasoning episodes. Order of presentation effects provided evidence of meta-analogical transfer based on transfer of predicate mappings (e.g., "successor to successor"). Experiments 2a and 2b reinforced these findings, demonstrating transfer when mappings for the first analogy were directly manipulated by having participants justify an answer to the first analogy. Experiment 3 demonstrated that a mapping of non-identical predicates ("successor to predecessor") can also be transferred. Meta-analogical transfer 3 Meta-Analogical Transfer: Transfer between Episodes of Analogical Reasoning Analogical reasoning is based on transferring information from a known situation (the source domain) to a situation requiring explaining (the target domain) by finding a similar system of relations that holds among the elements in each situation (Vosnaidou & Ortony, 1989). Such analogical reasoning has been suggested to underlie human thinking within domains as diverse as science (Oppenheimer, 1956), management (Isenberg, 1986), poetry (Holyoak, 1982), football recruiting, international relations (Gilovich, 1981), military command, chess, and fire fighting (Klein, 1986). One might ask, for example, whether a potential conflict involving the USA is another "Vietnam War" or another "World War II." Or in 1981, when Nancy Reagan was the "first lady" of the United States, who was the "first lady" of England? As Hofstadter (1985) pointed out, this question could have multiple answers: the Queen? Margaret Thatcher (at that time, the Prime Minister of the England)? Dennis Thatcher, who was Margaret Thatcher's husband? Hofstadter points out that Dennis Thatcher seems the most analogous answer because Nancy Reagan and Dennis Thatcher had the same relationship to political power. The ability of people to form abstract analogies is impressive (Gentner & Gentner, 1983; Gick & Holyoak, 1980, 1983; Keane, 1988). But once an episode of analogical reasoning is over, do the representations formed simply evaporate? Or, could the result of one episode of analogical reasoning be transferred to a new episode of analogical reasoning? Hofstadter (1995, p. 317) suggested that people can form meta-analogies, that is, analogies between analogies. He illustrated this concept with the following analogy: Doug: The Germans call a tortoise a Schildkröte  literally, "shield-toad". Carol: "Shield-toad"?! Come on! That's like calling an eagle a "feather-cow"! (Hofstadter, 1995, p. 304) In this meta-analogy, Doug points out that the German language suggests an analogy between a tortoise and a "shield-toad". Carol then ridicules this terminology, by drawing a new analogy between an eagle and a "feather-cow". In forming this new analogy Carol is transferring the analogical relationship between tortoise and "shield-toad" ("shield" has a good analogical correspondence to the shell that Meta-analogical transfer 4 covers a tortoise, but "toad" has a poor correspondence to its body) and using those correspondences to form a new analogy ("feathers" are what cover an eagle, but "cow" has little correspondence to an eagle's body). Another such meta-analogy can be illustrated using the previous example, that is, the analogy that Nancy Reagan was to America what Dennis Thatcher was to England. One might ask, who is to France as Albert Einstein is to Germany? In seeing Dennis Thatcher as analogous to Nancy Reagan the concept of "woman" in the source domain (i.e., Nancy Reagan being the first lady of America) is placed into correspondence with the concept of "man" in the target domain (i.e., Dennis Thatcher being the "first lady" of England). Such a correspondence can be described as a mapping of "woman to man" (in this paper the mapping of two concepts, X and Y, will be denoted as "X to Y"), which implies the mapping of "man to woman" as well. Accordingly, if people form a meta-analogy between the analogy that "Dennis Thatcher is the first lady of England" and the analogy problem "Who is the Albert Einstein of France", then they may transfer the mapping of "man to women". Thus people may be more likely to answer "Marie Curie". In a sense, meta-analogies are a form of learning from analogy, in that the mappings formed in one episode of analogical reasoning may assist with analogical reasoning in a new episode. A different form of learning from analogical reasoning was identified by Gick and Holyoak (1983), who found that making an analogy between two dissimilar stories (e.g., "The General" and "The Fire Chief", see Gick & Holyoak, 1983) resulted in the formation of an abstract schema that distilled the essence of the two stories (i.e., using converging weak forces in place of a single large force). People who formed such an abstract schema were better at solving a subsequent problem (i.e., The radiation problem of Duncker, 1945) that was analogous to both of the original stories, than if they had been given one analogue alone. Other studies have found similar advantages of forming a schema (Catrambone & Holyoak, 1989; Novick & Holyoak, 1991; Ross & Kennedy, 1990). But note that all the situations used in these experiments, including the source stories used to form the schema and the target problem, were analogous to each other. Individually, "The General" story is analogous to the radiation problem, as is the "Fire Chief" story. This design contrasts with meta-analogical transfer (i.e., transfer of elements of the first analogy to the second analogy through the formation of a meta-analogy). Meta-analogical transfer 5 In meta-analogical transfer, rather than all situations being analogous to each other, an analogy is found between the analogical relationships within each pair of situations comprising an analogical reasoning episode. Thus saying that "Nancy Reagan is the first lady of America" corresponds in no meaningful way to saying that "Einstein is the greatest physicist of Germany". The learning demonstrated in schema induction is learning of a refined source domain (i.e., the schema), rather than learning of a set of mappings that can potentially be applied to new analogies. Predicate Mappings as the Medium of Meta-Analogical Transfer Predicates represent the meaningful concepts (e.g., "man", "husband", "opposite-in") in a situation and are organized into propositions which consist of a predicate with a set of arguments (e.g., "man [Ronald Reagan]; "husband [Ronald Reagan, Nancy Reagan]"; "opposite-in [man, women, gender]"). Unless an argument to a predicate is itself a predicate, such arguments are referred to as objects. These propositions represent the structure of a situation by describing the relationship between parts of the situation. Thus predicate mappings are mappings between predicates in the source and target domains. For example, in the "Dennis Thatcher" analogy, mappings may be made of "husband to wife", "man to woman", and "opposite-in to opposite-in". Halford (1987) provides a useful taxonomy for the degree of complexity of predicate mappings: Element mappings (or attribute mappings, in the terminology of Gentner, 1983) map predicates that take one argument, such as, the proposition LAST(i) when describing the string of letters "kji". Relational mappings map predicates that have two arguments, such as, SUCCESSOR(k, j). System mappings involve predicates that have three arguments, such as, SEQUENCE(k, j, i). In this terminology an object mapping would be a mapping of two arguments of predicates, for example, mapping the letter i to some letter in another letter string. But transfer based on object mappings alone in the absence of predicate mappings would not be called analogical. This paper proposes that the medium of meta-analogical transfer is predicate mappings. This hypothesis is based on theories of analogical reasoning such as those of Gentner (1983) and Holyoak and Thagard (1989), which claim that people form coherent mappings between predicates in the source and target domains. For example, Gentner (1983) defines an analogy as an assertion that the relational Meta-analogical transfer 6 structure that normally applies in one domain can be applied in another domain. That is, the relations (which are represented by predicates) can be mapped between the domains. The emphasis on predicate mappings in models of analogical reasoning such as those of Gentner (1983) and Holyoak and Thagard (1989) accounts for our ability to form analogies when little semantic similarity exists between the source and target domains. In these models the mapping is assumed to be constrained by a pressure to maintain structural consistency. This constraint requires that mapped elements should play similar roles in both the source and target domains, and tend to map to only a single element in the other domain. To meet this requirement, a set of mutually constraining, coherent mappings are formed. As Holyoak and Thagard (1989, p. 295) state: "At the core of analogical thinking lies the process of mapping: the construction of orderly correspondences between elements of a source analog and those of a target." The flexibility of predicate mappings — allowing mappings of predicates with no semantic similarity — accounts for the power of analogical reasoning. In Genter's and Holyoak and Thagard's theories the analogy is a set of mappings. Thus if meta-analogical t