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C H A P T E R 6

Analogy

Keith J. Holyoak

Analogy is a special kind of similarity (seeGoldstone & Son, Chap. 2). Two situationsare analogous if they share a common pat-tern of relationships among their constituentelements even though the elements them-selves differ across the two situations. Typi-cally, one analog, termed the source or base, ismore familiar or better understood than thesecond analog, termed the target. This asym-metry in initial knowledge provides the ba-sis for analogical transfer, using the source togenerate inferences about the target. For ex-ample, Charles Darwin drew an analogy be-tween breeding programs used in agricultureto select more desirable plants and animalsand “natural selection” for new species. Thewell-understood source analog called atten-tion to the importance of variability in thepopulation as the basis for change in the dis-tribution of traits over successive generationsand raised a critical question about the tar-get analog: What plays the role of the farmerin natural selection? (Another analogy, be-tween Malthus’ theory of human populationgrowth and the competition of individuals ina species to survive and reproduce, providedDarwin’s answer to this question.) Analo-

gies have figured prominently in the historyof science (see Dunbar & Fugelsang, Chap.29) and mathematics (Pask, 2003) and are ofgeneral use in problem solving (see Novick &Bassok, Chap. 1 4). In legal reasoning, the useof relevant past cases (legal precedents) tohelp decide a new case is a formalized appli-cation of analogical reasoning (see Ellsworth,Chap. 28). Analogies can also function to in-fluence political beliefs (Blanchette & Dun-bar, 2001 ) and to sway emotions (Thagard& Shelley, 2001 ). Analogical reasoning goesbeyond the information initially given, usingsystematic connections between the sourceand target to generate plausible, althoughfallible, inferences about the target. Analogyis thus a form of inductive reasoning (seeSloman & Lagnado, Chap. 5).

Figure 6.1 sketches the major compo-nent processes in analogical transfer (seeCarbonell, 1983 ; Gentner, 1983 ; Gick &Holyoak, 1980, 1983 ; Novick & Holyoak,1991 ). Typically, a target situation servesas a retrieval cue for a potentially usefulsource analog. It is then necessary to es-tablish a mapping, or a set of systematiccorrespondences that serve to align the

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Figure 6.1 . Major components of analogicalreasoning.

elements of the source and target. On thebasis of the mapping, it is possible to de-rive new inferences about the target, therebyelaborating its representation. In the after-math of analogical reasoning about a pair ofcases, it is possible that some form of rela-tional generalization may take place, yieldinga more abstract schema for a class of situa-tions, of which the source and target are bothinstances. For example, Darwin’s use of anal-ogy to construct a theory of natural selectionultimately led to the generation of a more ab-stract schema for a selection theory, whichin turn helped to generate new specific the-ories in many fields, including economics,genetics, sociobiology, and artificial intelli-gence. Analogy is one mechanism for effect-ing conceptual change (see Chi & Ohlsson,Chap. 16).

A Capsule History

The history of the study of analogy in-cludes three interwoven streams of research,which respectively emphasize analogy in re-lation to psychometric measurement of in-

telligence, metaphor, and the representationof knowledge.

Psychometric Tradition

Work in the psychometric tradition focuseson four-term or “proportional” analogies inthe form A:B::C:D, such as HAND: FIN-GER :: FOOT: ?, where the problem is toinfer the missing D term (TOE) that is re-lated to C in the same way B is related toA (see Sternberg, Chap. 31 ). Thus A:B playsthe role of source analog and C:D plays therole of target. Proportional analogies werediscussed by Aristotle (see Hesse, 1966) andin the early decades of modern psychologybecame a centerpiece of efforts to defineand measure intelligence. Charles Spearman(1923 , 1927) argued that the best accountof observed individual differences in cogni-tive performance was based on a general org factor, with the remaining variance beingunique to the particular task. He reviewedseveral studies that revealed high correla-tions between performance in solving anal-ogy problems and the g factor. Spearman’sstudent John C. Raven (1938) developed theRaven’s Progressive Matrices Test (RPM),which requires selection of a geometric fig-ure to fill an empty cell in a two-dimensionalmatrix (typically 3 × 3) of such figures. Sim-ilar to a geometric proportional analogy, theRPM requires participants to extract and ap-ply information based on visuospatial rela-tions. (See Hunt, 1974 , and Carpenter, Just,& Shell, 1990, for analyses of strategies forsolving RPM problems.) The RPM proved tobe an especially pure measure of g.

Raymond Cattell (1971 ), another studentof Spearman, elaborated his mentor’s the-ory by distinguishing between two compo-nents of g: crystallized intelligence, which de-pends on previously learned information orskills, and fluid intelligence, which involvesreasoning with novel information. As a formof inductive reasoning, analogy would beexpected to require fluid intelligence. Cat-tell confirmed Spearman’s (1946) observa-tion that analogy tests and the RPM pro-vide sensitive measures of g, clarifying that


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Figure 6.2 . Multidimensional scaling solution based on intercorrelations among the Raven’sProgressive Matrices test, analogy tests, and other common tests of cognitive function. (From Snow,Kyllonen, & Marshalek, 1984 , p. 92 . Reprinted by permission.)

they primarily measure fluid intelligence(although verbal analogies based on diffi-cult vocabulary items also depend on crys-tallized intelligence). Figure 6.2 graphicallydepicts the centrality of RPM performancein a space defined by individual differencesin performance on various cognitive tasks.Note that numeric, verbal, and geometricanalogies cluster around the RPM at the cen-ter of the figure.

Because four-term analogies and the RPMare based on small numbers of relativelywell-specified elements and relations, it ispossible to manipulate the complexity ofsuch problems systematically and analyzeperformance (based on response latenciesand error rates) in terms of component

processes (e.g., Mulholland, Pellegrino, &Glaser, 1980; Sternberg, 1977). The earli-est computational models of analogy weredeveloped for four-term analogy problems(Evans, 1968; Reitman, 1965). The basiccomponents of these models were elabora-tions of those proposed by Spearman (1923),including encoding of the terms, accessinga relation between the A and B terms, andevoking a comparable relation between theC and D terms.

More recently, four-term analogy prob-lems and the RPM have figured promi-nently in neuropsychological and neu-roimaging studies of reasoning (e.g., Bunge,Wendelken, Badre & Wagner, 2004 ; Krogeret al., 2002 ; Luo et al., 2003 ; Prabhakaran



et al., 1997; Waltz et al., 1999; Whartonet al., 2000). Analogical reasoning dependson working memory (see Morrison, Chap.19). The neural basis of working memory in-cludes the dorsolateral prefrontal cortex, anarea of the brain that becomes increasinglyactivated as the complexity of the problem(measured in terms of number of relationsrelevant to the solution) increases. It hasbeen argued that this area underlies the fluidcomponent of Spearman’s g factor in intelli-gence (Duncan et al., 2000), and it plays animportant role in many reasoning tasks (seeGoel, Chap. 20).

Metaphor

Analogy is closely related to metaphor andrelated forms of symbolic expression thatarise in everyday language (e.g., “the eveningof life,” “the idea blossomed”), in literature(Holyoak, 1982), the arts, and cultural prac-tices such as ceremonies (see Holyoak &Thagard, 1995 , Chap. 9). Similar to anal-ogy in general, metaphors are characterizedby an asymmetry between target (conven-tionally termed “tenor”) and source (“ve-hicle”) domains (e.g., the target/tenor in“the evening of life” is life, which is un-derstood in terms of the source/vehicle oftime of day). In addition, a mapping (the“grounds” for the metaphor) connects thesource and target, allowing the domains tointeract to generate a new conceptualiza-tion (Black, 1962). Metaphors are a specialkind of analogy in that the source and tar-get domains are always semantically distant(Gentner, 1982 ; Gentner, Falkenhainer, &Skorstad, 1988), and the two domains areoften blended rather than simply mapped(e.g., in “the idea blossomed,” the targetis directly described in terms of an actionterm derived from the source). In addition,metaphors are often combined with othersymbolic “figures” – especially metonymy(substitution of an associated concept).For example, “sword” is a metonymic ex-pression for weaponry, derived from itsancient association as the prototypicalweapon – “Raising interests rates is the Fed-eral Reserve Board’s sword in the battle

against inflation” extends the metonymyinto metaphor.

Fauconnier and Turner (1998; Fauconnier,2001 ) analyzed complex conceptual blendsthat are akin to metaphor. A typical exam-ple is a description of the voyage of a mod-ern catamaran sailing from San Francisco toBoston that was attempting to beat the speedrecord set by a clipper ship that had sailedthe same route over a century earlier. Amagazine account written during the cata-maran’s voyage said the modern boat was“barely maintaining a 4 .5 day lead over theghost of the clipper Northern Light. . . . ” Fau-connier and Turner observed that the maga-zine writer was describing a “boat race” thatnever took place in any direct sense; rather,the writer was blending the separate voy-ages of the two ships into an imaginary race.The fact that such conceptual blends areso natural and easy to understand attests tothe fact that people can readily comprehendnovel metaphors.

Lakoff and Johnson (1980; also Lakoff &Turner, 1989) argued that much of humanexperience, especially its abstract aspects,is grasped in terms of broad conceptualmetaphors (e.g., events occurring in timeare understood by analogy to objects mov-ing in space). Time, for example, is under-stood in terms of objects in motion throughspace as in expressions such as “My birth-day is fast approaching” and “The time foraction has arrived.” (See Boroditsky, 2000,for evidence of how temporal metaphors in-fluence cognitive judgments.) As Lakoff andTurner (1989) pointed out, the course of alife is understood in terms of time in the solaryear (youth is springtime; old age is winter).Life is also conventionally conceptualized asa journey. Such conventional metaphors canstill be used in creative ways, as illustratedby Robert Frost’s famous poem, “The RoadNot Taken”:

Two roads diverged in a wood, and I –I took the one less traveled by,And that has made all the difference.

According to Lakoff and Turner, compre-hension of this passage depends on our im-plicit knowledge of the metaphor that life


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is a journey. This knowledge includes un-derstanding several interrelated correspon-dences (e.g., person is a traveler, purposesare destinations, actions are routes, diffi-culties in life are impediments to travel,counselors are guides, and progress is thedistance traveled).

Psychological research has focused ondemonstrations that metaphors are in-tegral to everyday language understand-ing (Glucksberg, Gildea, & Bookin, 1982 ;Keysar, 1989) and debate about whethermetaphor is better conceptualized as a kindof analogy (Wolff & Gentner, 2000) or akind of categorization (Glucksberg & Keysar,1990; Glucksberg, McClone, & Manfredi,1997). A likely resolution is that novelmetaphors are interpreted by much the sameprocess as analogies, whereas more conven-tional metaphors are interpreted as moregeneral schemas (Gentner, Bowdle, Wolff, &Boronat, 2001 ).

Knowledge Representation

The most important influence on analogyresearch in the cognitive science traditionhas been concerned with the representa-tion of knowledge within computational sys-tems. Many seminal ideas were developedby the philosopher Mary Hesse (1966), whowas in turn influenced by Aristotle’s dis-cussions of analogy in scientific classifica-tion and Black’s (1962) interactionist viewof metaphor. Hesse placed great stress onthe purpose of analogy as a tool for scien-tific discovery and conceptual change and onthe close connections between causal rela-tions and analogical mapping. In the 1970s,work in artificial intelligence and psychol-ogy focused on the representation of com-plex knowledge of the sort used in scientificreasoning, problem solving, story compre-hension, and other tasks that require struc-tured knowledge. A key aspect of structuredknowledge is that elements can be flexiblybound into the roles of relations. For exam-ple, “dog bit man” and “man bit dog” have thesame elements and the same relation, butthe role bindings have been reversed, radi-cally altering the meaning. How the mind

and brain accomplish role binding is thus acentral problem to be solved by any psycho-logical theory of structured knowledge, in-cluding any theory of analogy (see Doumas& Hummel, Chap. 4).

In the 1980s, a number of cognitive sci-entists recognized the centrality of analogyas a tool for discovery and its close connec-tion with theories of knowledge represen-tation. Winston (1980), guided by Minsky’s(1975) treatment of knowledge representa-tion, built a computer model of analogy thathighlighted the importance of causal rela-tions in guiding analogical inference. Otherresearchers in artificial intelligence also be-gan to consider the use of complex analogiesin reasoning and learning (Kolodner, 1983 ;Schank, 1982), leading to an approach to ar-tificial intelligence termed case-based reason-ing (see Kolodner, 1993).

Around 1980, two research projects inpsychology began to consider analogy inrelation to knowledge representation andeventually integrate computational model-ing with detailed experimental studies ofhuman analogical reasoning. Gentner (1982 ,1983 ; Gentner & Gentner, 1983) beganworking on mental models and analogy inscience. She emphasized that in analogy,the key similarities lie in the relations thathold within the domains (e.g., the flow ofelectrons in an electrical circuit is analog-ically similar to the flow of people in acrowded subway tunnel), rather than in fea-tures of individual objects (e.g., electronsdo not resemble people). Moreover, analog-ical similarities often depend on higher-orderrelations – relations between relations. For ex-ample, adding a resistor to a circuit causes adecrease in flow of electricity, just as adding anarrow gate in the subway tunnel would de-crease the rate at which people pass through(where causes is a higher-order relation). Inher structure-mapping theory, Gentner pro-posed that analogy entails finding a struc-tural alignment, or mapping, between do-mains. In this theory, alignment between tworepresentational structures is characterizedby structural parallelism (consistent, one-to-one correspondences between mappedelements) and systematicity – an implicit



preference for deep, interconnected systemsof relations governed by higher-order rela-tions, such as causal, mathematical, or func-tional relations.

Holyoak (1985 ; Gick & Holyoak, 1980,1983 ; Holyoak & Koh, 1987) focused on therole of analogy in problem solving with astrong concern for the role of pragmatics inanalogy – that is, how causal relations thatimpact current goals and context guide theinterpretation of an analogy. Holyoak andThagard (1989a, 1995) developed an ap-proach to analogy in which several factorswere viewed as jointly constraining analogi-cal reasoning. According to their multicon-straint theory, people tend to find mappingsthat maximize similarity of corresponding el-ements and relations, structural parallelism(i.e., isomorphism, defined by consistent,one-to-one correspondences), and prag-matic factors such as the importance of el-ements and relations for achieving a goal.Gick and Holyoak (1983) provided evidencethat analogy can furnish the seed for form-ing new relational categories by abstractingthe relational correspondences between ex-amples into a schema for a class of problems.Analogy was viewed as a central part of hu-man induction (Holland, Holyoak, Nisbett,& Thagard, 1986; see Sloman & Lagnado,Chap. 5) with close ties to other basicthinking processes, including causal infer-ence (see Buehner & Cheng, Chap. 7), cate-gorization (see Medin & Rips, Chap. 3), de-ductive reasoning (see Evans, Chap. 8),and problem solving (see Novick & Bassok,Chap. 1 4).

Analogical Reasoning: Overviewof Phenomena

This section provides an overview of themajor phenomena involving analogical rea-soning that have been established by em-pirical investigations. This review is orga-nized around the major components ofanalogy depicted in Figure 6.1 . These com-ponents are inherently interrelated, so theconnections among them are also discussed.

The retrieval and mapping components arefirst considered followed by inference andrelational generalization.

Retrieval and Mapping

a paradigm for investigating

analogical transfer

Gick and Holyoak (1980, 1983) introduceda general laboratory paradigm for investigat-ing analogical transfer in the context of prob-lem solving. The general approach was firstto provide people with a source analog inthe guise of some incidental context, suchas an experiment on “story memory.” Later,participants were asked to solve a problemthat was in fact analogous to the story theyhad studied earlier. The questions of cen-tral interest were (1 ) whether people wouldspontaneously notice the relevance of thesource analog and use it to solve the targetproblem, and (2) whether they could solvethe analogy once they were cued to considerthe source. Spontaneous transfer of the anal-ogous solution implies successful retrievaland mapping; cued transfer implies success-ful mapping once the need to retrieve thesource has been removed.

The source analog used by Gick andHolyoak (1980) was a story about a generalwho is trying to capture a fortress controlledby a dictator and needs to get his army tothe fortress at full strength. Because the en-tire army could not pass safely along any sin-gle road, the general sends his men in smallgroups down several roads simultaneously.Arriving at the same time, the groups jointogether and capture the fortress.

A few minutes after reading this storyunder instructions to read and remember it(along with two other irrelevant stories), par-ticipants were asked to solve a tumor prob-lem (Duncker, 1945), in which a doctor hasto figure out how to use rays to destroy astomach tumor without injuring the patientin the process. The crux of the problem isthat it seems that the rays will have the sameeffect on the healthy tissue as on the tumor –high intensity will destroy both, whereas lowintensity will destroy neither. The key issueis to determine how the rays can be made to


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impact the tumor selectively while sparingthe surrounding tissue. The source analog,if it can be retrieved and mapped, can beused to generate a “convergence” solution tothe tumor problem, one that parallels thegeneral’s military strategy: Instead of usinga single high-intensity ray, the doctor couldadminister several low-intensity rays at oncefrom different directions. In that way, eachray would be at low intensity along its path,and hence, harmless to the healthy tissue,but the effects of the rays would sum toachieve the effect of a high-intensity ray attheir focal point, the site of the tumor.

When Gick and Holyoak (1980) askedcollege students to solve the tumor problem,without a source analog, only about 10% ofthem produced the convergence solution.When the general story had been studied,but no hint to use it was given, only about20% of participants produced the conver-gence solution. In contrast, when the sameparticipants were then given a simple hintthat “you may find one of the stories you readearlier to be helpful in solving the problem,”about 75% succeeded in generating the anal-ogous convergence solution. In other words,people often fail to notice superficiallydissimilar source analogs that they couldreadily use.

This gap between the difficulty of re-trieving remote analogs and the relativeease of mapping them has been replicatedmany times, both with adults (Gentner,Rattermann, & Forbus, 1993 ; Holyoak &Koh, 1987; Spencer & Weisberg, 1986)and with young children (Chen, 1996;Holyoak, Junn, & Billman, 1984 ; Tunteler& Resing, 2002). When analogs mustbe cued from long-term memory, casesfrom a domain similar to that of thecue are retrieved much more readily thancases from remote domains (Keane, 1987;Seifert, McKoon, Abelson, & Ratcliff, 1986).For example, Keane (1987) measured re-trieval of a convergence analog to the tu-mor problem when the source analog wasstudied 1 to 3 days prior to presentation ofthe target radiation problem. Keane foundthat 88% of participants retrieved a sourceanalog from the same domain (a story about

a surgeon treating a brain tumor), whereasonly 1 2% retrieved a source from a remotedomain (the general story). This differencein ease of access was dissociable from theease of postaccess mapping and transfer be-cause the frequency of generating the con-vergence solution to the radiation prob-lem once the source analog was cued washigh and equal (about 86%), regardless ofwhether the source analog was from thesame or a different domain.

differential impact of similarity and

structure on retrieval versus mapping

The main empirical generalization concern-ing retrieval and mapping is that similar-ity of individual concepts in the analogshas a relatively greater impact on retrieval,whereas mapping is relatively more sensi-tive to relational correspondences (Gentneret al., 1993 ; Holyoak & Koh, 1987; Ross,1987, 1989). However, this dissociation isnot absolute. Watching the movie West SideStory for the first time is likely to trigger a re-minding of Shakespeare’s Romeo and Julietdespite the displacement of the charactersin the two works over centuries and conti-nents. The two stories both involve younglovers who suffer because of the disapprovalof their respective social groups, causing afalse report of death, which in turn leadsto tragedy. It is these structural parallels be-tween the two stories that make them anal-ogous rather than simply that both storiesinvolve a young man and woman, a disap-proval, a false report, and a tragedy.

Experimental work on story remindingconfirms the importance of structure, as wellas similarity of concepts, in retrieving analogsfrom memory. Wharton and his colleagues(Wharton et al., 1994 ; Wharton, Holyoak,& Lange, 1996) performed a series of exper-iments in which college students tried to findconnections between stories that overlappedin various ways in terms of the actors and ac-tions and the underlying themes. In a typicalexperiment, the students first studied abouta dozen “target” stories presented in the guiseof a study of story understanding. For exam-ple, one target story exemplified a theme of-ten called “sour grapes” after one of Aesop’s



fables. The theme in this story is that the pro-tagonist tries to achieve a goal, fails, and thenretroactively decides the goal had not reallybeen desirable after all. More specifically, theactions involved someone trying unsuccess-fully to get accepted to an Ivy League col-lege. After a delay, the students read a setof different cue stories and were asked towrite down any story or stories from thefirst session of which they were reminded.Some stories (far analogs) exemplified thesame theme, but with very different char-acters and actions (e.g., a “sour grapes” fairytale about a unicorn who tries to cross a riverbut is forced to turn back). Other storieswere far “disanalogs” formed by reorganizingthe characters and actions to represent a dis-tinctly different theme (e.g., “self-doubt” –the failure to achieve a goal leads the pro-tagonist to doubt his or her own ability ormerit). Thus, neither type of cue was simi-lar to the target story in terms of individualelements (characters and actions); however,the far analog maintained structural corre-spondences of higher-order causal relationswith the target story, whereas the far disana-log did not.

Besides varying the relation between thecue and target stories, Wharton et al. (1994)also varied the number of target stories thatwere in some way related to a single cue.When only one target story in a set had beenstudied (“singleton” condition), the proba-bility of reminding was about equal, regard-less of whether the cue was analogous to thetarget. However, when two target stories hadbeen studied (e.g., both “sour grapes” and“self-doubt,” forming a “competition” condi-tion), the analogous target was more likely tobe retrieved than the disanalogous one. Theadvantage of the far analog in the competi-tion condition was maintained even when aweek intervened between initial study of thetarget stories and presentation of the cue sto-ries (Wharton et al., 1996).

These results demonstrate that structuredoes influence analogical retrieval, but itsimpact is much more evident when multi-ple memory traces, each somewhat similarto the cue, must compete to be retrieved.Such retrieval competition is likely typical

of everyday analogical reminding. Other ev-idence indicates that having people generatecase examples, as opposed to simply askingthem to remember cases presented earlier,enhances structure-based access to sourceanalogs (Blanchette & Dunbar, 2000).

the “relational shift” in development

Retrieval is thus sensitive to structure anddirect similarity of concepts. Conversely,mapping is sensitive to direct similarity andstructure (e.g., Reed, 1987; Ross, 1989).Young children are particularly sensitiveto direct similarity of objects; when askedto identify corresponding elements in twoanalogs, their mappings are dominated byobject similarity when semantic and struc-tural constraints conflict (Gentner & Toupin,1986). Younger children are particularlylikely to map on the basis of object simi-larity when the relational response requiresintegration of multiple relations, and hence,is more dependent on working memory re-sources (Richland, Morrison, & Holyoak,2004). The developmental transition to-ward greater reliance on structure in map-ping has been termed the “relational shift”(Gentner & Rattermann, 1991 ). Greater sen-sitivity to relations with age appears to ariseowing to a combination of incremental ac-cretion of knowledge about relational con-cepts and stage-like increments in workingmemory capacity (Halford, 1993 ; Halford& Wilson, 1980). (For reviews of develop-mental research on analogy, see Goswami,1992 , 2001 ; Halford, Chap. 22 ; Holyoak &Thagard, 1995).

goal-directed mapping

Mapping is guided not only by relationalstructure and element similarity but also bythe goals of the analogist (Holyoak, 1985).People draw analogies not to find a pris-tine isomorphism for its own sake but tomake plausible inferences that will achievetheir goals. Particularly when the mappingis inherently ambiguous, the constraint ofpragmatic centrality – relevance to goals –is critical (Holyoak, 1985). Spellman andHolyoak (1996) investigated the impact of


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processing goals on the mappings gener-ated for inherently ambiguous analogies. Inone experiment, college students read twoscience fiction stories about countries ontwo planets. These countries were interre-lated by various economic and military al-liances. Participants first made judgmentsabout individual countries based on eithereconomic or military relationships and werethen asked mapping questions about whichcountries on one planet corresponded towhich on the other. Schematically, planet 1

included three countries, such that “Afflu”was economically richer than “Barebrute,”whereas the latter was militarily strongerthan “Compak.” Planet 2 included fourcountries, with “Grainwell” being richer than“Hungerall” and “Millpower” being strongerthan “Mightless.” The critical aspect of thisanalogy problem is that Barebrute (planet 1 )is both economically weak (like Hunger-all on planet 2) and militarily strong (likeMillpower) and therefore, has two compet-ing mappings that are equally supported bystructural and similarity constraints.

Spellman and Holyoak (1996) found thatparticipants whose processing goal led themto focus on economic relationships tendedto map Barebrute to Hungerall rather thanMillpower, whereas those whose process-ing goal led them to focus on militaryrelationships had the opposite preferredmapping. The variation in pragmatic cen-trality of the information thus served todecide between the competing mappings.One interpretation of such findings is thatpragmatically central propositions tend tobe considered earlier and more often thanthose that are less goal relevant and hence,dominate the mapping process (Hummel &Holyoak, 1997).

coherence in analogical mapping

The key idea of Holyoak and Thagard’s(1989a) multiconstraint theory of analogy isthat several different kinds of constraints –similarity, structure, and purpose – all in-teract to determine the optimal set of cor-respondences between source and target. Agood analogy is one that appears coherent in

the sense that multiple constraints convergeon a solution that satisfies as many differ-ent constraints as possible (Thagard, 2000).Everyday use of analogies depends on thehuman ability to find coherent mappings –even when source and target are complexand the mappings are ambiguous. For ex-ample, political debate often makes use ofanalogies between prior situations and somecurrent controversy (Blanchette & Dunbar,2001 , 2002). Ever since World War II, politi-cians in the United States and elsewherehave periodically argued that some militaryintervention was justified because the cur-rent situation was analogous to that lead-ing to World War II. A commonsensicalmental representation of World War II, thesource analog, amounts to a story figuringan evil villain, Hitler; misguided appeasers,such as Neville Chamberlain; and clear-sighted heroes, such as Winston Churchilland Franklin Delano Roosevelt. The coun-tries involved in World War II included thevillains, Germany and Japan; the victims,such as Austria, Czechoslovakia, and Poland;and the heroic defenders, notably Britain andthe United States.

A series of American presidents have usedthe World War II analog as part of theirargument for American military interven-tion abroad (see Khong, 1992). These in-clude Harry Truman (Korea, 1950), LyndonJohnson (Vietnam, 1965), George Bush se-nior (Kuwait and Iraq, 1991 ), and his sonGeorge W. Bush (Iraq, 2003). Analogies toWorld War II have also been used to sup-port less aggressive responses. Most notably,during the Cuban missile crisis of 1962 ,President John F. Kennedy decided againsta surprise attack on Cuba in part because hedid not want the United States to behave ina way that could be equated to Japan’s sur-prise attack on Pearl Harbor.

The World War II situation was, of course,very complex and is never likely to map per-fectly onto any new foreign policy problem.Nonetheless, by selectively focusing on goal-relevant aspects of the source and target andusing multiple constraints in combination,people can often find coherent mappings insituations of this sort. After the Iraqi invasion



of Kuwait in 1990, President GeorgeH. W. Bush argued that Saddam Hussein, theIraqi leader, was analogous to Adolf Hitlerand that the Persian Gulf crisis in general wasanalogous to events that had led to WorldWar II a half-century earlier. By drawing theanalogy between Hussein and Hitler, Pres-ident Bush encouraged a reasoning processthat led to the construction of a coherentsystem of roles for the players in the Gulf sit-uation. The popular understanding of WorldWar II provided the source, and analogicalmapping imposed a set of roles on the tar-get Gulf situation by selectively emphasizingthe most salient relational parallels betweenthe two situations. Once the analogical cor-respondences were established (with Iraqidentified as an expansionist dictatorship likeGermany, Kuwait as its first victim, SaudiArabia as the next potential victim, and theUnited States as the main defender of theGulf states), the clear analogical inferencewas that both self-interest and moral con-siderations required immediate military in-tervention by the United States. Aspects ofthe Persian Gulf situation that did not mapwell to World War II (e.g., lack of democracyin Kuwait) were pushed to the background.

Of course, the analogy between the twosituations was by no means perfect. Simi-larity at the object level favored mappingthe United States of 1991 to the UnitedStates of World War II simply because itwas the same country, which would in turnsupport mapping Bush to President Roo-sevelt. However, the United States did notenter World War II until it was bombedby Japan, well after Hitler had marchedthrough much of Europe. One might there-fore argue that the United States of 1991

mapped to Great Britain of World War II andthat Bush mapped to Winston Churchill, theBritish Prime Minister (because Bush, sim-ilar to Churchill, led his nation and West-ern allies in early opposition to aggression).These conflicting pressures made the map-pings ambiguous. However, the pressure tomaintain structural consistency implies thatpeople who mapped the United States toBritain should also tend to map Bush toChurchill, whereas those who mapped the

United States to the United States shouldinstead map Bush to Roosevelt.

During the first 2 days of the U.S.-ledcounterattack against the Iraqi invasion ofKuwait, Spellman and Holyoak (1992) askeda group of American undergraduates a fewquestions to find out how they interpretedthe analogy between the then-current situ-ation in the Persian Gulf and World War II.The undergraduates were asked to sup-pose that Saddam Hussein was analogousto Hitler. Regardless of whether they be-lieved the analogy was appropriate, theywere then asked to write down the mostnatural match in the World War II situationfor Iraq, the United States, Kuwait, SaudiArabia, and George Bush. For those stu-dents who gave evidence that they knew thebasic facts about World War II, the major-ity produced mappings that fell into one oftwo patterns. Those students who mappedthe United States to itself also mappedBush to Roosevelt; these same students alsotended to map Saudi Arabia to Great Britain.Other students, in contrast, mapped theUnited States to Great Britain and Bush toChurchill, which in turn (so as to maintainone-to-one correspondences) forced SaudiArabia to map to some country other thanBritain. The mapping for Kuwait (which didnot depend on the choice of mappings forBush, the United States, or Saudi Arabia)was usually to one or two of the early vic-tims of Germany in World War II (usuallyAustria or Poland).

The analogy between the Persian Gulf sit-uation and World War II thus generated a“bistable” mapping: People tended to pro-vide mappings based on either of two coher-ent but mutually incompatible sets of corre-spondences. Spellman and Holyoak (1992)went on to perform a second study, using adifferent group of undergraduates, to showthat people’s preferred mappings could bepushed around by manipulating their knowl-edge of the source analog, World War II.Because many undergraduates were lackingin knowledge about the major participantsand events in World War II, it proved pos-sible to “guide” them to one or the othermapping pattern by having them first read a


analogy 1 2 7

slightly biased summary of events in WorldWar II. The various summaries were all his-torically “correct,” in the sense of providingonly information taken directly from historybooks, but each contained slightly differ-ent information and emphasized differentpoints. Each summary began with an iden-tical passage about Hitler’s acquisition ofAustria, Czechoslovakia, and Poland and theefforts by Britain and France to stop him.The versions then diverged. Some versionswent on to emphasize the personal role ofChurchill and the national role of Britain;other versions placed greater emphasis onwhat Roosevelt and the United States didto further the war effort. After reading oneof these summaries of World War II, the un-dergraduates were asked the same mappingquestions as had been used in the previousstudy. The same bistable mapping patternsemerged as before, but this time the sum-maries influenced which of the two coher-ent patterns of responses students tendedto give. People who read a “Churchill” ver-sion tended to map Bush to Churchill andthe United States to Great Britain, whereasthose who read a “Roosevelt” version tendedto map Bush to Roosevelt and the UnitedStates to the United States. It thus ap-pears that even when an analogy is messyand ambiguous, the constraints on analog-ical coherence produce predictable inter-pretations of how the source and targetfit together.

Achieving analogical coherence in map-ping does not, of course, guarantee that thesource will provide a clear and compellingbasis for planning a course of action to dealwith the target situation. In 1991 , PresidentBush considered Hussein enough of a Hitlerto justify intervention in Kuwait but notenough of one to warrant his removal frompower in Iraq. A decade later his son, Presi-dent George W. Bush, reinvoked the WorldWar II analogy to justify a preemptive inva-sion of Iraq itself. Bush claimed (falsely, aswas later revealed) that Hussein was acquir-ing biological and perhaps nuclear weaponsthat posed an imminent threat to the UnitedStates and its allies. Historical analogies canbe used to obfuscate as well as to illuminate.

RelationalMatchFeatural

Match

TargetObject

Figure 6.3 . An example of a pair of picturesused in studies of analogical mapping witharrows added to indicate featural and relationalresponses. (From Tohill & Holyoak, 2000, p. 31 .Reprinted by permission.)

working memory in analogical mapping

Analogical reasoning, because it depends onmanipulating structured representations ofknowledge, would be expected to make crit-ical use of working memory. The role ofworking memory in analogy has been ex-plored using a picture-mapping paradigm in-troduced by Markman and Gentner (1993).An example of stimuli similar to those theyused is shown in Figure 6.3 . In their exper-iments, college students were asked to ex-amine the two pictures and then decide (forthis hypothetical example) what object inthe bottom picture best goes with the manin the top picture. When this single map-ping is considered in isolation, people oftenindicate that the boy in the bottom picturegoes with the man in the top picture basedon perceptual and semantic similarity ofthese elements. However, when people areasked to match not just one object but three(e.g., the man, dog, and the tree in the top



picture to objects in the bottom picture),they are led to build an integrated represen-tation of the relations among the objects andof higher-order relations between relations.In the top picture, a man is unsuccessfullytrying to restrain a dog, which then chasesthe cat. In the bottom picture, the tree is un-successful in restraining the dog, which thenchases the boy. Based on these multiple in-teracting relations, the preferred match tothe man in the top picture is not the boy inthe lower scene but the tree. Consequently,people who map three objects at once aremore likely to map the man to the tree onthe basis of their similar relational roles thanare people who map the man alone.

Whereas Markman and Gentner (1993)showed that the number of objects to bemapped influences the balance betweenthe impact of element similarity versus re-lational structure, other studies using thepicture-mapping paradigm have demon-strated that manipulations that constrictworking memory resources have a similarimpact. Waltz, Lau, Grewal, and Holyoak(2000) asked college students to map pic-tures while performing a secondary taskdesigned to tax working memory (e.g., gen-erating random digits). Adding a dual task di-minished relational responses and increasedsimilarity-based responses (see Morrison,Chap. 19). A manipulation that increasespeople’s anxiety level (performing mathe-matical calculations under speed pressureprior to the mapping task) yielded a sim-ilar shift in mapping responses (Tohill &Holyoak, 2000). Most dramatically, degen-eration of the frontal lobes radically impairsrelation-based mapping (Morrison et al.,2004). In related work using complex storyanalogs, Krawczyk, Holyoak, and Hummel(2004) demonstrated that mappings (and in-ferences) based on element similarity ver-sus relational structure were made aboutequally often when the element similaritieswere salient and the relational structure washighly complex. All these findings supportthe hypothesis that mapping on the basis ofrelations requires adequate working mem-ory to represent and manipulate role bind-ings (Hummel & Holyoak, 1997).

Inference and Relational Generalization

copy with substitution and generation

Analogical inference – using a source analogto form a new conjecture, whether it be astep toward solving a math problem (Reed,Dempster, & Ettinger, 1985 ; see Novick& Bassok, Chap. 1 4), a scientific hypoth-esis (see Dunbar & Fugelsang, Chap. 29),a diagnosis for puzzling medical symptoms(see Patel, Arocha, & Zhang, Chap. 30),or a basis for deciding a legal case (seeEllsworth, Chap. 28) – is the fundamentalpurpose of analogical reasoning. Mappingserves to highlight correspondences betweenthe source and target, including “alignabledifferences” (Markman & Gentner, 1993) –the distinct but corresponding elements ofthe two analogs. These correspondences pro-vide the input to an inference engine thatgenerates new target propositions. The ba-sic form of analogical inference has beencalled “copy with substitution and genera-tion” (CWSG; Holyoak et al., 1994). CWSGinvolves constructing target analogs of un-mapped source propositions by substitutingthe corresponding target element, if known,for each source element, and if no corre-sponding target element exists, postulatingone as needed. This procedure gives rise totwo important corollaries concerning infer-ence errors. First, if critical elements are dif-ficult to map (e.g., because of strong repre-sentational asymmetries such as those thathinder mapping a discrete set of elementsto a continuous variable; Bassok & Holyoak,1989; Bassok & Olseth, 1995), then no in-ferences can be constructed. Second, if ele-ments are mismapped, predictable inferenceerrors will result (Holyoak et al., 1994 ; Reed,1987).

All major computational models of ana-logical inference use some variant of CWSG(e.g., Falkenhainer et al., 1989; Halford et al.,1994 ; Hofstadter & Mitchell, 1994 ; Holyoaket al., 1994 ; Hummel & Holyoak, 2003 ;Keane & Brayshaw, 1988; Kokinov & Petrov,2001 ). CWSG is critically dependent onvariable binding and mapping; hence, mod-els that lack these key computational prop-erties (e.g., traditional connectionist models)


analogy 1 2 9

fail to capture even the most basic as-pects of analogical inference (see Doumas &Hummel, Chap. 4).

Athough all analogy models use someform of CWSG, additional constraintson this inference mechanism are critical(Clement & Gentner, 1991 ; Holyoak et al.,1994 ; Markman, 1997). If CWSG wereunconstrained, then any unmapped sourceproposition would generate an inferenceabout the target. Such a loose criterion forinference generation would lead to ram-pant errors whenever the source was notisomorphic to a subset of the target, andsuch isomorphism will virtually never holdfor problems of realistic complexity. Sev-eral constraints on CWSG were demon-strated in a study by Lassaline (1996; alsosee Clement & Gentner, 1991 ; Spellman& Holyoak, 1996). Lassaline had collegestudents read analogs describing proper-ties of hypothetical animals and then ratevarious possible target inferences for theprobability that the conclusion would betrue given the information in the premise.Participants rated potential inferences asmore probable when the source and tar-get analogs shared more attributes, andhence, mapped more strongly. In addition,their ratings were sensitive to structuraland pragmatic constraints. The presenceof a higher-order linking relation in thesource made an inference more credible. Forexample, if the source and target animalswere both described as having an acutesense of smell, and the source animal wassaid to have a weak immune system that“develops before” its acute sense of smell,then the inference that the target animal alsohas a weak immune system would be bol-stered relative to stating only that the sourceanimal had an acute sense of smell “and”a weak immune system. The benefit con-veyed by the higher-order relation was in-creased if the relation was explicitly causal(e.g., in the source animal, a weak immunesystem “causes” its acute sense of smell),rather than less clearly causal (“developsbefore”). (See Hummel & Holyoak, 2003 ,for a simulation of this and other inferenceresults using a CWSG algorithm.)

An important question is when analogi-cal inferences are made and how inferencesgenerated by CWSG relate to facts about thetarget analog that are stated directly. Oneextreme possibility is that people only makeanalogical inferences when instructed to doso and that inferences are carefully “marked”as such so they will never be confused withknown facts about the target. At the otherextreme, it is possible that some analogi-cal inferences are triggered when the tar-get is first processed (given that the sourcehas been activated) and that such inferencesare then integrated with prior knowledgeof the target. One paradigm for address-ing this issue is based on testing for false“recognition” of potential inferences in asubsequent memory test. The logic of therecognition paradigm (Bransford, Barclay, &Franks, 1972) is that if an inference has beenmade and integrated with the rest of thetarget analog, then later the reasoner willfalsely believe that the inference had beendirectly presented.

Early work by Schustack and Anderson(1979) provided evidence that people some-times falsely report that analogical infer-ences were actually presented as facts.Blanchette and Dunbar (2002) performeda series of experiments designed to assesswhen analogical inferences are made. Theyhad college students (in Canada) read a textdescribing a current political issue, possiblelegalization of marijuana use, which servedas the target analog. Immediately afterward,half the students read, “The situation withmarijuana can be compared to . . . ”, followedby an additional text describing the periodearly in the twentieth century when alco-hol use was prohibited. Importantly, the stu-dents in the analogy condition were not toldhow prohibition mapped onto the marijuanadebate, nor were they asked to draw any in-ferences. After a delay (1 week in one ex-periment, 1 5 minutes in another), the stu-dents were given a list of sentences and wereasked to decide whether each sentence hadactually been presented in the text aboutmarijuana use. The critical items were sen-tences such as “The government could set upagencies to control the quality and take over


1 30 the cambridge handbook of thinking and reasoning

the distribution of marijuana.” These sen-tences had never been presented; however,they could be generated as analogical infer-ences by CWSG based on a parallel state-ment contained in the source analog (“Thegovernment set up agencies to control thequality and take over the distribution of al-cohol”). Blanchette and Dunbar found thatstudents in the analogy condition said “yes”to analogical inferences about 50% of thetime, whereas control subjects who had notread the source analog about prohibition said“yes” only about 25% of the time. This ten-dency to falsely “recognize” analogical infer-ences that had never been read was obtainedboth after long and short delays and withboth familiar and less familiar materials.

It thus appears that when people noticethe connection between a source and target,and they are sufficiently engaged in an effortto understand the target situation, analogi-cal inferences will be generated by CWSGand then integrated with prior knowledge ofthe target. At least sometimes, an analogicalinference becomes accepted as a stated fact.This result obviously has important impli-cations for understanding analogical reason-ing, such as its potential for use as a toolfor persuasion.

relational generalization

In addition to generating local inferencesabout the target by CWSG, analogical rea-soning can give rise to relational general-izations – abstract schemas that establishan explicit representation of the common-alities between the source and the target.Comparison of multiple analogs can resultin the induction of a schema, which inturn will facilitate subsequent transfer toadditional analogs. The induction of suchschemas has been demonstrated in bothadults (Catrambone & Holyoak, 1989; Gick& Holyoak, 1983 ; Loewenstein, Thompson,& Gentner, 1999; Ross & Kennedy, 1990)and young children (Brown, Kane, & Echols,1986; Chen & Daehler, 1989; Holyoak et al.,1984 ; Kotovsky & Gentner, 1996). Peopleare able to induce schemas by comparingjust two analogs to one another (Gick &

Holyoak, 1983). Indeed, people will formschemas simply as a side effect of applyingone solved source problem to an unsolvedtarget problem (Novick & Holyoak, 1991 ;Ross & Kennedy, 1990).

In the case of problem schemas, moreeffective schemas are formed when thegoal-relevant relations are the focus ratherthan incidental details (Brown et al., 1986;Brown, Kane, & Long, 1989; Gick &Holyoak, 1983). In general, any kind of pro-cessing that helps people focus on the under-lying causal structure of the analogs, therebyencouraging learning of more effective prob-lem schemas, will improve subsequent trans-fer to new problems. For example, Gickand Holyoak (1983) found that inductionof a “convergence” schema from two dis-parate analogs was facilitated when eachstory stated the underlying solution prin-ciple abstractly: “If you need a large forceto accomplish some purpose, but are pre-vented from applying such a force directly,many smaller forces applied simultaneouslyfrom different directions may work just aswell.” In some circumstances, transfer canalso be improved by having the reasonergenerate a problem analogous to an initialexample (Bernardo, 2001 ). Other work hasshown that abstract diagrams that highlightthe basic idea of using multiple convergingforces can aid in schema induction and sub-sequent transfer (Beveridge & Parkins, 1987;Gick & Holyoak, 1983) – especially whenthe diagram uses motion cues to convey per-ception of forces acting on a central target(Pedone, Hummel, & Holyoak, 2001 ; seeFigure 6.4 , top).

Although two examples can suffice to es-tablish a useful schema, people are able toincrementally develop increasingly abstractschemas as additional examples are provided(Brown et al., 1986, 1989; Catrambone &Holyoak, 1989). However, even with mul-tiple examples that allow novices to startforming schemas, people may still fail totransfer the analogous solution to a prob-lem drawn from a different domain if asubstantial delay intervenes or if the con-text is changed (Spencer & Weisberg, 1986).Nonetheless, as novices continue to develop


analogy 1 31

Figure 6.4. Sequence of diagrams used to convey the convergence schema byperceived motion. Top: sequence illustrating convergence (arrows appear tomove inward in II–IV). Bottom: control sequence in which arrows divergeinstead of converge (arrows appear to move outward in II–IV). (From Pedone,Holyoak, & Hummel, 2001 , p. 217. Reprinted by permission.)

more powerful schemas, long-term transferin an altered context can be dramaticallyimproved (Barnett & Koslowski, 2002). Forexample, Catrambone and Holyoak (1989)gave college students a total of three con-vergence analogs to study, compare, andsolve. The students were first asked a seriesof detailed questions designed to encouragethem to focus on the abstract structure com-mon to two of the analogs. After this ab-straction training, the students were askedto solve another analog from a third do-main (not the tumor problem), after whichthey were told the convergence solution toit (which most students were able to gen-erate themselves). Finally, 1 week later, thestudents returned to participate in a dif-ferent experiment. After the other experi-ment was completed, they were given thetumor problem to solve. More than 80%of participants came up with the converg-ing rays solution without any hint. As thenovice becomes an expert, the emergingschema becomes increasingly accessible andis triggered by novel problems that share itsstructure. Deeper similarities have been con-

structed between analogous situations thatfit the schema. As schemas are acquiredfrom examples, they in turn guide futuremappings and inferences (Bassok, Wu, &Olseth, 1995).

Computational Models of Analogy

From its inception, work on analogyin relation to knowledge representationhas involved the development of detailedcomputational models of the various com-ponents of analogical reasoning typically fo-cusing on the central process of structuremapping. The most influential early modelsincluded SME (Structure Mapping Engine;Falkenhainer, Forbus, & Gentner, 1989),ACME (Analogical Mapping by ConstraintSatisfaction; Holyoak & Thagard, 1989a),IAM (Incremental Analogy Model; Keane &Brayshaw, 1988), and Copycat (Hofstadter& Mitchell, 1994). More recently, modelsof analogy have been developed based onknowledge representations constrained byneural mechanisms (Hummel & Holyoak,



1992). These efforts included an approachbased on the use of tensor products for vari-able binding, the STAR model (StructuredTensor Analogical Reasoning; Halford et al.,1994 ; see Halford, Chap. 22), and anotherbased on neural synchrony, the LISA model(Learning and Inference with Schemas andAnalogies; Hummel & Holyoak, 1997, 2003 ;see Doumas & Hummel, Chap. 4). (For abrief overview of computational models ofanalogy, see French, 2002 .) Three models aresketched to illustrate the general nature ofcomputational approaches to analogy.

Structure Mapping Engine (SME)

SME (Falkenhainer et al., 1989) illustrateshow analogical mapping can be performedby algorithms based on partial graph match-ing. The basic knowledge representation forthe inputs is based on a notation in the styleof predicate calculus. If one takes a simpleexample based on the World War II analogyas it was used by President George Bush in1991 , a fragment might look like

SOURCE:Fuhrer-of (Hitler, Germany)occupy (Germany, Austria)evil (Hitler)cause [evil (Hitler), occupy (Germany,

Austria)]prime-minister-of (Churchill, Great

Britain)cause [occupy (Germany, Austria), coun-

terattack (Churchill, Hitler)]TARGET:president-of (Hussein, Iraq)invade (Iraq, Kuwait)evil (Hussein)cause [evil (Hussein), invade (Iraq,

Kuwait)]president-of (Bush, United States)

SME distinguishes objects (role fillers,such as “Hitler”), attributes (one-place pred-icates, such as “evil” with its single role filler),first-order relations (multiplace predicates,such as “occupy” with its two role fillers), andhigher-order relations (those such as “cause”that take at least one first-order relation as arole filler). As illustrated in Figure 6.5 , the

predicate-calculus notation is equivalent to agraph structure. An analogical mapping canthen be viewed as a set of correspondencesbetween partially matching graph structures.

The heart of the SME algorithm is a pro-cedure for finding graph matches that sat-isfy certain criteria. The algorithm operatesin three stages, progressing in a “local-to-global” direction. First, SME proposes lo-cal matches between all identical predicatesand their associated role fillers. It is as-sumed similar predicates (e.g., “Fuhrer-of”and “president-of”; “occupy” and “invade”)are first transformed into more general pred-icates (e.g.,“leader-of”; “attack”) that reveala hidden identity. (In practice, the program-mer must make the required substitutionsso similar but nonidentical predicates can bematched.) The resulting matches are typi-cally inconsistent in that one element in thesource may match multiple elements in thetarget (e.g., Hitler might match either Hus-sein or Bush because all are “leaders”). Sec-ond, the resulting local matches are inte-grated into structurally consistent clusters or“kernels” (e.g., the possible match betweenHitler and Bush is consistent with that be-tween Germany and the United States, andso these matches would form part of a sin-gle kernel). Third, the kernels are mergedinto a small number of sets that are max-imal in size (i.e., that include matches be-tween the greatest number of nodes in thetwo graphs), while maintaining correspon-dences that are structurally consistent andone to one. SME then ranks the result-ing sets of mappings by a structural eval-uation metric that favors “deep” mappings(ones that include correspondences betweenhigher-order relations). For our example,the optimal set will respectively map Hitler,Germany, Churchill, and Great Britain toHussein, Iraq, Bush, and the United Statesbecause of the support provided by the map-ping between the higher-order “cause” rela-tions involving “occupy/invade.” Using thisoptimal mapping, SME applies a CWSG al-gorithm to generate inferences about thetarget based on unmapped propositions inthe source. Here, the final “cause” relation


analogy 1 33

Figure 6.5 . SME’s graphical representation of a source and target analog.

in the source will yield the analogical infer-ence, cause [attack (Iraq, Kuwait), counter-attack (Bush, Hussein)].

SME thus models the mapping and in-ference components of analogical reason-ing. A companion model, MACFAC (“ManyAre Called but Few Are Chosen”; Forbus,Gentner, & Law, 1995) deals with the ini-tial retrieval of a source analog from long-term memory. MACFAC has an initial stage(“many are called”) in which analogs are rep-resented by content vectors, which code therelative number of occurrences of a partic-

ular predicate in the corresponding struc-tured representation. (Content vectors arecomputed automatically from the underly-ing structural representations.) The contentvector for the target is then matched to vec-tors for all analogs stored in memory, andthe dot product for each analog pair is cal-culated as an index of similarity. The sourceanalog with the highest dot product, plusother stored analogs with relatively high dotproducts, are marked as retrieved. In its sec-ond stage, MACFAC uses SME to assessthe degree of the structural overlap between



the target and each possible source, allowingthe program to identify a smaller number ofpotential sources that have the highest de-grees of structural parallelism with the target(“few are chosen”). As the content vectorsused in the first stage of MACFAC do notcode role bindings, the model provides aqualitative account of why the retrieval stageof analogy is less sensitive to structure thanis the mapping stage.

Analogical Mapping by ConstraintSatisfaction (ACME)

The ACME model (Holyoak, Novick, &Melz, 1994 ; Holyoak & Thagard, 1989a) wasdirectly influenced by connectionist mod-els based on parallel constraint satisfac-tion (Rumelhart, Smolensky, McClelland,& Hinton, 1986; see Doumas & Hummel,Chap. 4). ACME takes as input symbolicrepresentations of the source and targetanalogs in essentially the same form as thoseused in SME. However, whereas SME fo-cuses on structural constraints, ACME in-stantiates a multiconstraint theory in whichstructural, semantic, and pragmatic con-straints interact to determine the optimalmapping. ACME accepts a numeric codefor degree of similarity between predicates,which it uses as a constraint on mapping.Thus, ACME, unlike SME, can match simi-lar predicates (e.g., “occupy” and “invade”)without explicitly recoding them as iden-tical. In addition, ACME accepts a nu-meric code for the pragmatic importanceof a possible mapping, which is also usedas a constraint.

ACME is based on a constraint satis-faction algorithm, which proceeds in threesteps. First, a connectionist “mapping net-work” is constructed in which the units rep-resent hypotheses about possible elementmappings and the links represent specific in-stantiations of the general constraints (Fig-ure 6.6). Second, an interactive-activationalgorithm operates to “settle” the map-ping network in order to identify the setof correspondences that collectively repre-sent the “optimal” mapping between theanalogs. Any constraint may be locally vio-

lated to establish optimal global coherence.Third, if the model is being used to gener-ate inferences and correspondences, CWSGis applied to generate inferences basedon the correspondences identified in thesecond step.

ACME has a companion model, ARCS(Analog Retrieval by Constraint Satis-faction; Thagard, Holyoak, Nelson, &Gochfeld, 1990) that models analog re-trieval. Analogs in long-term memory areconnected within a semantic network (seeMedin & Rips, Chap. 3); this network ofconcepts provides the initial basis by whicha target analog activates potential sourceanalogs. Those analogs in memory that areidentified as having semantic links to the tar-get (i.e., those that share similar concepts)then participate in an ACME-like con-straint satisfaction process to select the opti-mal source. The constraint network formedby ARCS is restricted to those conceptsin each analog that have semantic links;hence, ARCS shows less sensitivity to struc-ture in retrieval than does ACME in map-ping. Because constraint satisfaction algo-rithms are inherently competitive, ARCScan model the finding that analogical ac-cess is more sensitive to structure when sim-ilar source analogs in long-term memorycompete to be retrieved (Wharton et al.,1994 , 1996).

Learning and Inference with Schemasand Analogies (LISA)

Similar to ACME, the LISA model(Hummel & Holyoak, 1997, 2003) isbased on the principles of the multicon-straint theory of analogy; unlike ACME,LISA operates within psychologically andneurally realistic constraints on workingmemory (see Doumas & Hummel, Chap. 4 ;Morrison, Chap. 19). The models discussedpreviously include at most localist rep-resentations of the meaning of concepts(e.g., a semantic network in the case ofARCS), and most of their processing isperformed on propositional representationsunaccompanied by any more detailed levelof conceptual representation (e.g., neither


analogy 1 35

PURPOSE SIMILARITY

Saddam

Hitler

president-of

Führer-of

invade

occupy

Kuwait

Austria

Kuwait

Germany

invade

Führer-of

Iraq

Hitler

Iraq

Germany

Iraq

Austria

Saddam

Germany

president-of

occupy

Figure 6.6. A constraint-satisfaction network in ACME.

ACME nor SME includes any represen-tation of the meaning of concepts). LISAalso goes beyond previous models in thatit provides a unified account of all themajor components of analogical reasoning(retrieval, mapping, inference, and re-lational generalization).

LISA represents propositions using a hi-erarchy of distributed and localist units (seeFigure 4 .1 in Doumas & Hummel, Chap. 4).LISA includes both a long-term memoryfor propositions and concept meanings anda limited-capacity working memory. LISA’sworking memory representation, which usesneural synchrony to encode role-filler bind-ings, provides a natural account of the ca-pacity limits of working memory because itis only possible to have a finite number ofbindings simultaneously active and mutuallyout of synchrony.

Analog retrieval is accomplished as a formof guided pattern matching. Propositions in atarget analog generate synchronized patterns

of activation on the semantic units, which inturn activate propositions in potential sourceanalogs residing in long-term memory. Theresulting coactivity of source and target el-ements, augmented with a capacity to learnwhich structures in the target were coactivewith which in the source, serves as the basisfor analogical mapping. LISA includes a setof mapping connections between units of thesame type (e.g., object, predicate) in sepa-rate analogs. These connections grow when-ever the corresponding units are active si-multaneously and thereby permit LISA tolearn the correspondences between struc-tures in separate analogs. They also permitcorrespondences learned early in mapping toinfluence the correspondences learned later.Augmented with a simple algorithm for self-supervised learning, the mapping algorithmserves as the basis for analogical inferenceby CWSG. Finally, augmented with a sim-ple algorithm for intersection discovery, self-supervised relational learning serves as the



basis for schema induction. LISA has beenused to simulate a wide range of data onanalogical reasoning (Hummel & Holyoak,1997, 2003), including both behavioraland neuropsychological studies (Morrisonet al., 2004).

Conclusions and Future Directions

When we think analogically, we do muchmore than just compare two analogs basedon obvious similarities between their el-ements. Rather, analogical reasoning is acomplex process of retrieving structuredknowledge from long-term memory, repre-senting and manipulating role-filler bind-ings in working memory, performing self-supervised learning to form new inferences,and finding structured intersections betweenanalogs to form new abstract schemas. Theentire process is governed by the core con-straints provided by isomorphism, similarityof elements, and the goals of the reasoner(Holyoak & Thagard, 1989a). These con-straints apply in all components of analog-ical reasoning: retrieval, mapping, inference,and relational generalization. When analogsare retrieved from memory, the constraint ofelement similarity plays a large role, but rela-tional structure is also important – especiallywhen multiple source analogs similar to thetarget are competing to be selected. Formapping, structure is the most importantconstraint but requires adequate workingmemory resources; similarity and purposealso contribute. The success of analogical in-ference ultimately depends on whether thepurpose of the analogy is achieved, but satis-fying this constraint is intimately connectedwith the structural relations between theanalogs. Finally, relational generalization oc-curs when schemas are formed from thesource and target to capture those structuralpatterns in the analogs that are most rele-vant to the reasoner’s purpose in exploitingthe analogy.

Several current research directions arelikely to continue to develop. Computa-tional models of analogy, such as LISA(Hummel & Holyoak, 1997, 2003), have

begun to connect behavioral work on anal-ogy with research in cognitive neuroscience(Morrison et al., 2004). We already havesome knowledge of the general neural cir-cuits that underlie analogy and other formsof reasoning (see Goel, Chap. 20). Asmore sophisticated noninvasive neuroimag-ing methodologies are developed, it shouldbecome possible to test detailed hypothe-ses about the neural mechanisms underly-ing analogy, such as those based on temporalproperties of neural systems.

Most research and modeling in the fieldof analogy has emphasized quasilinguisticknowledge representations, but there is goodreason to believe that reasoning in generalhas close connections to perception (e.g.,Pedone et al., 2001 ). Perception providesan important starting point for grounding atleast some “higher” cognitive representations(Barsalou, 1999). Some progress has beenmade in integrating analogy with perception.For example, the LISA model has been aug-mented with a Metric Array Module (MAM;Hummel & Holyoak, 2001 ), which providesspecialized processing of metric informationat a level of abstraction applicable to bothperception and quasispatial concepts. How-ever, models of analogy have generally failedto address evidence that the difficulty ofsolving problems and transferring solutionmethods to isomorphic problems is depen-dent on the difficulty of perceptually encod-ing key relations. The ease of solving appar-ently isomorphic problems (e.g., isomorphsof the well-known Tower of Hanoi) can varyenormously, depending on perceptual cues(Kotovsky & Simon, 1990; see Novick & Bas-sok, Chap. 1 4).

More generally, models of analogy havenot been well integrated with models ofproblem solving (see Novick & Bassok,Chap. 1 4), even though analogy clearly af-fords an important mechanism for solvingproblems. In its general form, problem solv-ing requires sequencing multiple operators,establishing subgoals, and using combina-tions of rules to solve related but noni-somorphic problems. These basic require-ments are beyond the capabilities of vir-tually all computational models of analogy(but see Holyoak & Thagard, 1989b, for


analogy 1 37

an early although limited effort to inte-grate analogy within a rule-based problem-solving system). The most successful modelsof human problem solving have been formu-lated as production systems (see Lovett &Anderson, Chap. 1 7), and Salvucci and An-derson (2001 ) developed a model of anal-ogy based on the ACT-R production system.However, this model is unable to solve re-liably any analogy that requires integrationof multiple relations – a class that includesanalogies within the grasp of young children(Halford, 1993 ; Richland et al., 2004 ; seeHalford, Chap. 22). The integration of anal-ogy models with models of general problemsolving remains an important research goal.

Perhaps the most serious limitation ofcurrent computational models of analogy isthat their knowledge representations mustbe hand-coded by the modeler, whereas hu-man knowledge representations are formedautonomously. Closely related to the chal-lenge of avoiding hand-coding of represen-tations is the need to flexibly rerepresentknowledge to render potential analogies per-spicuous. Concepts often have a close con-ceptual relationship with more complex re-lational forms (e.g., Jackendoff, 1983). Forexample, causative verbs such as lift (e.g.,“John lifted the hammer”) have very simi-lar meanings to structures based on an ex-plicit higher-order relation, cause (e.g., “Johncaused the hammer to rise”). In such cases,the causative verb serves as a “chunked” rep-resentation of a more elaborate predicate-argument structure. People are able to “see”analogies even when the analogs have verydifferent linguistic forms (e.g., “John liftedthe hammer in order to strike the nail” mightbe mapped onto “The Federal Reserve usedan increase in interest rates as a tool in itsefforts to drive down inflation”). A deeperunderstanding of human knowledge repre-sentation is a prerequisite for a complete the-ory of analogical reasoning.

Acknowledgments

Preparation of this chapter was supported bygrants R305H030141 from the Institute ofEducation Sciences and SES-0080375 from

the National Science Foundation. KevinDunbar and Robert Morrison provided valu-able comments on an earlier draft.

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