the difficulty of the linda conjunction problem can be attributed to its simultaneous concrete and...

The Difficulty of the Linda Conjunction Problem Can BeAttributed to Its Simultaneous Concrete and UnnaturalRepresentation, and Not to Conversational Implicature

SEAN DONOVAN AND SEYMOUR EPSTEIN

University of Massachusetts at Amherst

Received: November 8, 1995; revised: May 1, 1996; accepted: June 14, 1996

To determine whether the high rate of conjunction errors (CEs) to the notorious Lindaproblem can be explained by the violation of implicit conversational rules, rather than byits concrete-unnatural representation, as proposed by cognitive-experiential self-theory,participants were given completely disclosing information. Although this procedure,directed toward a rational mode of information processing, reduced CEs, a majority ofparticipants continued to make CEs. A graded series of problems designed to activatelatent, intuitive knowledge, but not a procedure designed to provide additional informationof a rational nature, additionally reduced CEs. The implications of the findings arediscussed with respect to two independent, parallel modes of information processing:experiential-intuitive and rational-analytic. In certain situations the outcome of the experi-ential-intuitive mode is more compelling than that of the rational-analytical mode, evenwhen the latter is equally accessible. Our findings indicate that the resistance of the Lindaproblem to a probabilistic solution is even greater than previously suspected.r 1997

Academic Press

The study reported here concerns the notorious Linda conjunction problem.Linda is described as a bright, 31-year-old, single woman who majored inphilosophy in college and was concerned about social justice. Participants areasked to rank order the probability that Linda is a feminist, a bank teller, and afeminist and a bank teller, among other alternatives. Although, according to theconjunction rule, the joint occurrence of two events cannot be more likely than theoccurrence of either of the separate events, Tversky and Kahneman (1983) foundthat 85% of their participants ranked the conjunction of bank teller and feminist asmore probable than the constituent of bank teller. This statistical violation, which

Preparation of this manuscript and the research reported in it were supported by NIMH researchgrant MH 01293 and by NIMHResearch Scientist Award 5 KO5MH00363 to Epstein.We express ourappreciation to Alice Epstein for her constructive comments on a preliminary version of this report.Address reprint requests and correspondence to Seymour Epstein, Department of Psychology,University of Massachusetts, Amherst, MA 01003. E-mail: [email protected].

JOURNAL OF EXPERIMENTAL SOCIAL PSYCHOLOGY33, 1–20 (1997)ARTICLE NO. JS961309

1

0022-1031/97 $25.00Copyrightr 1997 by Academic Press

All rights of reproduction in any form reserved.

JESP 1309@spsun1/disk3/CLS_jrnl/GRP_jesp/JOB_jesp97ps/DIV_320z01 root

No. of Pages—20 First page no.—1 Last page no.—20

Tversky and Kahneman termed a conjunction fallacy, and which we shall refer toas a conjunction error (CE), has been replicated by Tversky and Kahneman andothers many times over.What has intrigued researchers about the Linda problem is that although the

conjunction rule is one of the simplest and most fundamental principles ofprobability theory (Tversky & Kahneman, 1983), the vast majority of respon-dents, including many who are statistically sophisticated, violate it when it ispresented in the form of the Linda problem (Tversky & Kahneman, 1983).Moreover, a variety of remediation procedures have been found, at best, to beonly modestly effective. No matter what procedures Tversky & Kahneman (1983)used to decrease CEs to theintactLinda problem, a surprisingly large number ofpeople continued to make CEs.Because CEs to the very simple intact Linda problem occur at a surprisingly

high rate and are extremely resistant to elimination, some have doubted that theLinda problem is a true conjunction problem. They argue, for example, that theconjunction rule is applicable to frequency but not to probability estimates, or thatit is applicable to samples of people but not to attributes of a single individual(e.g., Cosmides & Tooby, in press; Fiedler, 1988; Gigerenzer, 1991). In support ofthe former view, a study is widely cited by Fiedler (1988) who reportedimpressive differences as a function of whether his dependent variable consistedof frequency or probability estimates. However, his results cannot be interpretedunambiguously because he used different response formats for the two versions ofthe Linda problem, one that required ranking probabilities (the usual procedure)and the other that required exact frequency estimates. Moreover, the frequencyestimates required some very strange decisions, such as estimating the number ofpeople out of 100 who ‘‘work in a bookstore and take Yoga classes.’’ In a secondexperiment, Fiedler used probability ratings for both versions, making the resultsinapplicable to the more common ranking procedure.In two series of studies using a ranking procedure and including only the

conjunction and its constituents, we obtained equally high rates of CEs inprobability estimates of attributes in a single person and in frequency estimates ina sample of people (Epstein & Donovan, 1995; Epstein & Pacini, 1995). The issueabout multiple attributes in single individuals is further clarified by a simplethought experiment. Imagine that you had to bet on whether a person you did notknow possessed one or two attributes. Would you prefer to bet that the person wasabove average in height and had green eyes or that he or she was just aboveaverage in height? Whereas it is true that a person either has or does not havecertain attributes, your level of certainty in the absence of prior knowledge isreasonably less for a combination of attributes than for a single constituent. It maybe concluded that the Linda problem is a viable conjunction problem.

Explanations of the Diffıculty of the Linda Problem

Tversky and Kahneman’s (1983) explanation of why the Linda problem elicitsan unduly high rate of CEs is that it is processed by ‘‘natural assessments.’’ They

2 DONOVAN AND EPSTEIN


define natural assessments as ‘‘assessments that are routinely carried out as part ofthe perception of events and the comprehension of messages. Such naturalassessments include computations of similarity and representativeness, attribu-tions of causality and evaluations of the availability of associations and exem-plars’’ (1983, p. 294). For the Linda problem, Tversky and Kahneman believe anatural assessment consists of thinking in terms of the representativeness heuris-tic. As a bank teller and a feminist are more representative of a political activist(Linda) than a bank teller alone, the conjunction is judged more likely than theconstituent.Our explanation is similar to that of Tversky and Kahneman, but is more

articulated and within the context of a general theory of personality, cognitive-experiential self-theory (CEST). According to CEST, people process informationin two parallel systems, rational and experiential. The rational system is aconscious, deliberative, analytical, primarily verbal system with a very briefevolutionary history. The experiential system is a preconscious, automatic,intuitive, primarily imagistic system with a very long evolution history. Heuristicprocessing is its natural mode of operation. It is further assumed that the systemsare interactive, and that behavior varies along a continuum reflecting their relativeinfluence. It is also assumed that the experiential system is the default option, andthat everyday behavior is primarily determined by it (for a more detailedcomparison of the attributes of the two systems, see Table 1). The followingprinciples of the experiential system are particularly relevant to conjunctionerrors: the natural mode of operation of the experiential system corresponds toheuristic processing; the experiential system is a concretive system; experientialprocessing is intrinsically highly compelling and can override rational processing,even when the latter is equally accessible.We also differ from Tversky and Kahneman in that we operationally define and

measure natural processing. Also, we, unlike they, account for the difficulty of theLinda problem by its simultaneous location on two critical dimensions: natural–unnatural and concrete–abstract (Epstein, Denes-Raj, & Pacini, 1995). We as-sume that concrete-unnatural problems are inherently difficult to solve because ofthe operating principles of the experiential system, as noted above.Let us now more specifically consider the interaction of the two dimensions. If

a conjunction problem is concrete and natural (shortly to be operationallydefined), such as one that requires a decision about whether it is more likely that aperson will win two lotteries or just one of them, almost no one makes CEs(Epstein, Denes-Raj, & Pacini, 1995). In such a case, the concrete representationfacilitates a correct solution as it engages the experiential system, which processesthe information in the usual way of responding to such problems (lotteries) inprobabilistic terms based on past experience with similar problems. Correctsolutions for concrete-natural problems can therefore be obtained by processinginformation in the experiential mode, in the rational mode, or by the use of bothmodes. The important point is that the two modes operate in harmony, rather thanconflictually. The opposite condition exists for concrete-unnatural problems, such

3LINDA PROBLEM


as the Linda problem. Here, the concrete representation interferes with a correctsolution, as natural processing based on past experience with similar problemsfavors a nonprobabilistic solution. The general principle can be stated thatconcrete representations facilitate correct solutions when problems require natu-ral responses and impede correct solutions when problems require unnaturalresponses. After the fact, this may appear self evident. However, it is noteworthythat both Tversky and Kahneman (1983) and Yates and Carlson (1986) expresssurprise that although concrete problems are generally more easily solved thanabstract problems, exactly the opposite is the case for the Linda problem, andneither ventures an explanation of why this is so.We define concrete as a discrete object or event that can be directly experienced

or imagined. We define abstract as a generalization that cannot be directlyexperienced or imagined, but must be inferred. A specific house is concrete,whereas the category ‘‘house’’ is abstract. The Linda problem is concrete becauseit describes a specific event. The same problem presented in the form of algebraicsymbols would be abstract.We define natural reasoning as the type of reasoning that is most characteristi-

TABLE 1COMPARISON OF THEEXPERIENTIAL AND RATIONAL SYSTEMS

Experiential system Rational system

1. Holistic 1. Analytic2. Affective: Pleasure-pain oriented (what

feels good)2. Logical: Reason oriented (what is sensible)

3. Associationistic connections 3. Logical connections4. Behavior mediated by ‘‘vibes’’ from past

experiences4. Behavior mediated by conscious appraisal

of events5. Encodes reality in concrete images, meta-

phors, & narratives5. Encodes reality in abstract symbols, words,

& numbers6. More rapid processing: Oriented toward

immediate action6. Slower processing: Oriented toward

delayed action7. Slower to change: Changes with repetitive

or intense experience7. Changes more rapidly: Changes with speed

of thought8. More crudely differentiated: Broad general-

ization gradient; stereotypical thinking8. More highly differentiated

9. More crudely integrated: Dissociative, emo-tional complexes; context-specific pro-cessing

9. More highly integrated: Cross-context pro-cessing

10. Experienced passively and preconsciously:We are seized by our emotions

10. Experienced actively and consciously: Weare in control of our thoughts

11. Self-evidently valid: ‘‘Experiencing isbelieving’’

11. Requires justification via logic & evidence

Note.From Epstein, S., Cognitive-experiential self-theory: An integrative theory of personality. InR. C. Curtis (Ed.),The relational self: Theoretical convergences in psychoanalysis and socialpsychology.New York: The Guilford Press. Copyright 1991 by Guilford Press. Adapted by permis-sion.



cally exhibited in response to a particular problem. Correspondingly, we define anatural problem as one that can be solved by natural reasoning. By this definition,the Linda problem is unnatural because it typically elicits representativenessreasoning and a correct solution requires statistical reasoning. By the same token,a lottery conjunction problem is natural because it typically elicits probabilisticreasoning, which is the kind of reasoning required for a correct solution.The dimensions we have proposed are useful for research purposes because

their operationalization can be objectively and independently accomplished fromthe measurement of CEs. For example, the naturalness of a conjunction problem,like the Linda problem, can be quantified by obtaining from a representativesample of respondents the reasons for their responses, which can then beclassified by judges according to whether they are indications of representative-ness, statistical, or miscellaneous reasoning. The Linda problem has been foundby this procedure to be highly unnatural, as it evokes a high percentage ofrepresentativeness reasoning, whereas the correct solution requires statisticalreasoning. Employing this procedure, we have demonstrated that other variablesthat are known to influence CE rates, including problem order, compatibility ofconstituent events, within- versus between-protagonist context, and statisticalsophistication, cannot account for the robust influence of the combination of theconcrete–abstract and natural–unnatural dimensions (Epstein, Denes-Raj, & Pacini,1995). It should be noted that there is nothing circular about our operationaldefinition of naturalness, as the kind of reasoning people engage in and thecorrectness of their responses are not equivalent. That is, it is possible to obtainincorrect responses despite the correct kind of reasoning (e.g., errors in probabilis-tic reasoning), and, fortuitously, to obtain correct responses despite an incorrecttype of reasoning (e.g., representativeness reasoning for conjunction problems),and, in fact, both of these kinds of responses occur for the Linda vignette far moreoften than has been generally realized (Epstein, Denes-Raj, & Pacini, 1995).As, according to our definitions, the natural–unnatural and the concrete–

abstract divisions are independent, it follows that there are four kinds ofconjunction problems that can be investigated, and about which predictions canbe made according to CEST: concrete-natural problems (e.g., lottery problemsthat elicit primarily statistical reasoning), concrete-unnatural problems (e.g., theLinda problem, that elicits primarily representativeness reasoning), abstract-natural problems (e.g., problems presented in the form of algebraic symbols, thatelicit primarily statistical reasoning), and abstract-unnatural problems (e.g.,problems in theoretical physics that require creative solutions).Although it has been established that many variables known to affect CE rates

are relatively minor compared to the combined influence of the dimensions ofconcrete–abstract and natural–unnatural (Epstein, Denes-Raj, & Pacini, 1995),there is a potentially highly important variable that has not yet been adequatelyexamined, namely ‘‘conversational implicature’’ (Grice, 1975; Schwarz, 1994;Sperber & Wilson, 1986). It has been argued by those who have appliedconversational implicature to studies on decisional processes that the widespread

5LINDA PROBLEM


finding of an astonishing degree of human fallibility in reasoning can be attributedto the violation of implicit conversational rules by experimenters, leading partici-pants to misapply their efforts (see review in Schwarz, 1994). The implication ofthis for the Linda problem is that participants are misled into reasoning in arepresentativeness rather than in a statistical manner because of the context inwhich the conjunction problem is presented.As Sperber andWilson (1986) note, afundamental axiom in normal communication is that ‘‘communicated informationcomes with a guarantee of relevance’’ (p. vi). This rule is clearly violated in theLinda problem, with its emphasis on irrelevant personality information.If, in fact, misleading communication can account for the high rate of CEs to

the Linda problem, it follows that the natural-processing view of Tversky andKahneman and our elaboration of that view with an emphasis on the importanceof simultaneous concrete and unnatural representation is superfluous. On theother hand, if it can be demonstrated that a high rate of CEs persists despitenullification of the misleading information, it would not only indicate thatTversky and Kahneman’s and our views are viable, but it would provideimpressive evidence that the outcome of processing in the experiential mode is socompelling with reference to the Linda problem as to override rational-analyticalconsiderations.The most important purpose of the present experiment was to determine

whether CEs to the Linda problem could be virtually eliminated by presenting theproblem in a manner that does not violate implicit rules of normal conversation.To this end, we provided completely disclosing information. Participants wereadvised to regard all problems as essentially statistical in nature and to not bemisled by extraneous information. The reason for including the extraneousinformation was legitimized by presenting it as a challenge to the respondents’ability to identify and discount its influence. In summary, the instructionsremoved any implicit deception about the examiner’s intention of how theproblem should be solved and provided a reasonable explanation of why theextraneous information was included. Stated most simply, we ‘‘blew the cover’’on the Linda problem.So far as we can determine, despite increasingly desperate attempts by Tversky

and Kahneman and others to reduce CEs to the Linda problem (none of whichhave come close to eliminating CEs to the intact problem) nothing as extreme asour procedure has been attempted. Although Dulany and Hilton (1991) claim tohave virtually eliminated CEs by applying Gricean principles, they did so bysimply redefining CEs, maintaining that the endorsement of the conjunction asmore probable than its constituents should not be regarded as a CE unlessparticipants violate principles of reasoning with respect to their own view aboutthe nature of the problem. This, of course, begs the issue, as it fails to explain whyparticipants frame the problem in the manner they do. The current research shouldhelp clarify this issue because it explicitly directs the participants to regard theLinda problem as probabilistic.It should be noted that we were not at all interested in whether providing



disclosing information could significantly reduce CEs. For good reason based onpast research, we assumed it would. It is well known that one reason people makeCEs is that, despite formally knowing the conjunction rule, they do not think of itin the context of the Linda problem (Epstein, Denes-Raj, & Pacini, 1995; Tversky& Kahneman, 1983). Thus, informing participants to regard the problem asessentially probabilistic should alert at least this group to consider the relevanceof the conjunction rule, which should result in a significant reduction in CEs,which would tell us nothing that is not already known. Our interest was indetermining whether disclosing information couldvirtually eliminateCEs. Forthis reason, the use of a control group was superfluous.The present research had several secondary aims. One was to determine the

relative effect of providing additional information, beyond the disclosing informa-tion, to the rational and experiential systems. It was anticipated that additionalinformation directed at the rational system would have less of an influence thanrelevant experience directed at the experiential system, as the latter is less likely tobe redundant with the influence of the disclosing information. To determinewhether a different kind of information directed at the rational system wouldimprove performance beyond any improvement produced by the disclosinginstructions, we provided one group of participants with the correct principle,among others. At the very least, this information should serve as a reminder of theconjunction rule for those who know it formally. On the assumption that mostpeople who explicitly know the conjunction rule would be sufficiently remindedto use it by the disclosing instructions, alone, we predicted that including theconjunction rule would have little, if any, additional effect on reducing CEs.To examine the effect of providing information to the experiential system, we

presented another group of participants with a series of concrete-natural problemspreceding the Linda problem. This procedure supplies no information thatparticipants do not already possess, but rather provides anexperienceassumed toactivate latent intuitive knowledge that they do possess. If participants do nothave such knowledge, they cannot benefit from exposure to the concrete-naturalvignettes, but, as we have demonstrated elsewhere (Epstein, Denes-Raj, & Pacini,1995), virtually everyone has intuitive knowledge of the conjunction rule. Thus,this procedure should facilitate the application of the conjunction rule to the Lindaproblem by activating intuitive knowledge. Accordingly, we predicted that thisprocedure, directed at the experiential system, would produce a significantdecrease in CEs beyond that produced by the disclosing instructions, directed atthe rational system.Another secondary purpose of the study was to verify the explanatory power of

a double classification system based on the dimensions of abstract versus concreteand natural versus unnatural. The importance of the interaction of these twodimensions, as already noted, follows from the operating principles of theexperiential system, as proposed in CEST. The double classification receivedsome support in previous research that examined a few relevant stimuli (Epstein,Denes-Raj, & Pacini, 1995). In the present study we wished to test the generality

7LINDA PROBLEM


of the previous findings by examining a broader sample of stimuli speciallyconstructed for the purpose. For reasons already indicated, we predicted thatconcrete-natural problems (e.g., lotteries, horse races) would have relatively highrates of solution, and that concrete-unnatural problems (e.g., the Linda problem),and abstract-natural problems (e.g., formal conjunction problems) would haveconsiderably lower rates of solution. We were unable to make a prediction aboutthe order of the latter two categories, as there is no a priori basis for determiningthe relative influence of abstractness and unnaturalness.A final interest was to verify two findings from our previous research that have

important methodological implications for research on conjunction problems.One concerns the importance of establishing the reasoning process underlying theovert responses. As already noted, our previous research suggested that thecustomary practice of inferring mode of reasoning from the correctness ofresponses to the Linda problem leads to serious errors in interpretation, as theLinda problem elicits a high rate of correct responses fortuitously, i.e., for wrong(representativeness) reasons. Because of its importance, we considered it desir-able to establish the replicability and generality of this observation acrossdifferent samples and experimental procedures. To this end, we required onegroup of participants in the present experiment to write the reasons for theirresponses and another to select the correct rule from a number of possibilities,thereby allowing us to determine whether the latter procedure, which is mucheasier to score, produces equivalent results to the free-response procedure. Weanticipated that a considerable proportion of correct responses to the Lindaproblem would be obtained for wrong (representativeness) reasons, and that thestrength of coherent relations between variables would be increased when rightresponses for wrong reasons were treated as wrong.The other methodological issue concerns the measurement of sophistication

about the conjunction rule. Typically, in previous research, sophistication hasbeen inferred from exposure to statistics courses, and the results have beeninconsistent across studies (Tversky & Kahneman, 1983). In our own research,we have measured statistical sophistication directly by giving a brief test ofquantitative conjunction problems (Epstein, Denes-Raj, & Pacini, 1995). Wefound that many who had a statistics course could not solve such problems,whereas several who had not taken a statistics course could. Moreover, there wasa significant positive relation between performance on the Linda problem withscores on the statistics test, but not with exposure to statistics courses. It remainsto be seen whether our findings will be consistent across studies in a way that thefindings based on inferences of sophistication from exposure to statistics courseshave not been.

METHODParticipantsOne-hundred-forty-four college undergraduates (33 men, 111 women) participated in the experi-

ment for course credit. Equal numbers (n5 48) were randomly assigned to three experimental



conditions (Reasons, Principles, and Control), and half in each condition were randomly assigned toreceiving the Linda vignette first and half last.

Materials and ProcedureBooklets were constructed that contained seven vignettes, one on each page (see Appendix for

transcriptions of vignettes). In one set of booklets, the first vignette was abstract-natural, requiringparticipants to judge which was more likely to occur, ‘‘B,’’ an unlikely event, or ‘‘Aand B,’’ acombination of a likely and an unlikely event. This problem provides a direct test of explicitknowledge of the conjunction rule. We presented it to half the participants first in order to obtain ameasure of knowledge of the conjunction rule uninfluenced by exposure to previous vignettes.Following the abstract problem, six concrete vignettes were presented in order of increasingunnaturalness, with the Linda vignette presented last. The remaining booklets were identical exceptthat the Linda vignette appeared first (before the abstract vignette, which was now second instead offirst).In order to construct concrete-unnatural vignettes that were less unnatural than the Linda vignette,

we included less personality information in one case, and in the other we included a constituent of theconjunction that was unrelated to personality, namely, eye color.The cover of each booklet contained the following information, designed to remove the implicit

deception in the standard presentation of the problem, and to assure participants that they would bepresented only with simple conjunction problems that they should readily be able to solve:

What is the chance of two events happening compared to one? This is a problem thatarises in many simple situations in everyday life, such as when people bet on two lotteries.Most of the time, the answers to such problems are self-evident, and even a child can solvethem. However, psychologists have found that by adding extra information that distractspeople from the simple statistical solution that is required, or by presenting the problem in aform that suggests it is a different kind of problem, people often fail to recognize that theproblem simply requires a comparison between the likelihood of two events occurringcompared to one.Please consider all of the problems that follow as essentially statistical problems. Some of

the problems test your ability to find a disguised statistical problem. Your job is to avoidbeing distracted by extraneous information or by the form in which the problem is presented.Before giving your answers to any problem, pleaseread the entire information, including

all the questions, on the problem.Once you turn a page after responding to a problem, pleasedo not turn backto change your answers. Youmay, however, turn back to this page to reviewthe instructions at any time.

After reading the questions and other information following a vignette, participants indicatedwhether they thought the conjunction or its less likely constituent was more likely to occur, andanswered other questions, depending on to which of three conditions they had been assigned. If theywere assigned to the Principles condition, they endorsed what they believed was the correct principlefrom among three principles that were listed. If they were assigned to the Reasons condition, theywrote the reasons for their responses. If they were assigned to the control condition, they did nothingbeyond the basic ranking.The special instructions for the Principles condition were as follows:

Which of the following do you believe states the correct principle for solving theproblem? (Check one.)____ The probability of two events both occurring is greater than the probability of either ofthe single events occurring.____ The probability of two events both occurring is somewhere between the probabilitiesof the single events occurring.

9LINDA PROBLEM


____ The probability of two events both occurring is less than the probability of either of thesingle events occurring.

The special instructions for the Reasons group were as follows: ‘‘State the reason for your responsein the space below.’’Participants were asked to indicate whether they had taken a statistics course. The final task was a

quantitative conjunction problem that required participants to compute the joint occurrence of twoevents, one with a 1 in 10 and theother with a 1 in 2chance of occurring. The purpose of this problemwas to determine whether they could apply the conjunction rule to a simple quantitative problem.Participants were examined in groups of 30–40. They read the instructions to themselves while the

experimenter read them aloud. When the experiment was over, they were given experimental credit,debriefed, and thanked for their participation.

RESULTS

The Linda Vignette

In this section, we analyze only responses to the Linda vignette.Conjunction errors.Despite informing participants to be on guard against

statistical problems embedded in nonstatistical contexts, 57% across all condi-tions made CEs. This is but a modest improvement over the 68 and 78% figuresobtained in two of our previous studies that used standard instructions andparticipants from the same population (Epstein, Denes-Raj, & Pacini, 1995).An analysis of variance (ANOVA) of dichotomous data (Lumney, 1970), with

CEs as the dependent variable, and order (Linda first vs Linda last) andexperimental conditions (standard, principles, reasons) as the independent vari-ables, revealed a significant effect for order, and no other effect was significant. Insupport of prediction, participants presented with the Linda vignette following thegraded series of increasingly unnatural vignettes made significantly fewer CEs(49%) than participants presented with the Linda vignette first (65%),F(1, 138)54.24,p, .05.Conjunction errors and kinds of reasoning reported in the Reasons condition.

In this section, only the data in the Reasons condition are considered. Two judgesindependently sorted the reasons participants gave in three categories: statistical,representativeness, and miscellaneous. Inter-rater agreement was 86%; a thirdjudge resolved differences. Of the 48 responses, only one was scored as miscella-neous. Among the remaining 47 participants in the Reasons condition, 83%engaged in representativeness reasoning and 17% in statistical reasoning. Compar-ing the 83% who reported representativeness reasoning with the 62% who madeCEs, it is apparent that a substantial number of participants in the Reasons conditionfortuitously avoidedCEs despite engaging in representativeness reasoning.A x2 analysis revealed that type of reasoning and the production of CEs were

significantly associated,x2 (1) 5 6.65, p , .01. Of the 39 participants whoengaged in representativeness reasoning, 72% made CEs and 28% fortuitouslyavoided CEs; of the 8 participants who engaged in statistical reasoning, 25%made CEs and 75% avoided CEs. These findings indicate that despite theexpected association between kind of reasoning and incidence of CEs, there aremany who made correct responses for wrong reasons. It follows that the



customary procedure of inferring mode of reasoning from correctness of re-sponses is unwarranted, and, relatedly, that it is important to determine thereasons for responses.When the data in the Reasons condition were reclassified by treating right

responses for representativeness reasons as wrong responses, the percentage ofincorrect responses in the Reasons condition increased from 64 to 87%, suggest-ing that the 57% of CEs reported for the entire group of 144 participantsconsiderably underestimates the degree of incorrect (representativeness) reason-ing. Thus, as extensive as CEs have been reported to be in the past, our findingssuggest that the rate of representativeness processing inferred from such results isan underestimate.Conjunction errors and the kind of statistical principles endorsed in the

Principles condition.Among the 48 participants in the Principles condition, 52%endorsed the correct statistical principle (the conjunction rule), 29% endorsed thestatement that the likelihood of the conjunction falls between its constituents, and19% endorsed the statement that the conjunction is more likely than either of itsconstituents. Not surprisingly, there were more respondents in the Principlescondition who selected the correct principle (52%) than there were those in theReasons condition who thought of it by themselves (13%). This indicates thatendorsement of principles that are presented cannot be used as a substitute forfreely generated responses. It also indicates that presenting the correct principleprovides a reminder to those who know it, but might not have otherwise thoughtof it.A x2 analysis revealed a significant relation between endorsed principles and

CEs,x2 (1)5 3.93,p, .05. Seventy percent of those who endorsed an incorrectstatistical principle made CEs as compared to only 40% of those who endorsedthe correct principle. Endorsing the correct principle, as might be expected, isdirectly associated with producing correct responses. However, as in the Reasonscondition, the relation is far from perfect. Thirty percent of the participants whoendorsed an incorrect statistical principle avoided CEs, and 40% of those whoendorsed the correct statistical principle made CEs.Apparently, many participantswere inconsistent between the principle they endorsed and their ranking of theconjunction. It is noteworthy that many who ‘‘knew better’’ did not apply theirabstract knowledge to their responses to the Linda problem.

The Other Six Vignettes

Conjunction errors as a function of type of vignette.Conjunction errors wereinvestigated as a function of the three types of vignettes: abstract-natural,concrete-natural, and concrete-unnatural. The vignette that referred to algebraicletters was the sole representative of the abstract-natural category. The threevignettes that involved betting (lottery, auto race, and horse race) represented theconcrete-natural category, and the Mary and Paul vignettes represented theconcrete-unnatural category. The Linda vignette was not included in the initialanalysis of the concrete-unnatural category because it already had been analyzed

11LINDA PROBLEM


in a separate design. Combining the vignettes within the concrete-natural andwithin the concrete-unnatural categories was justified by their coefficients ofinternal consistency (coefficienta) of .72 and .70, respectively.A complication in comparing the different categories of vignettes with each

other is that a balanced design with respect to order had to be sacrificed in order topresent the vignettes in the order of increasing elicitation of representativenessreasoning. This confound introduces a conservative error, as it operates againstthe prediction that concrete-unnatural vignettes will elicit more CEs than concrete-natural vignettes. Since the concrete, unnatural vignettes were presented after thenatural vignettes, they could profit from exposure to the natural vignettes, acondition demonstrated to significantly reduce CEs to the Linda vignette.A repeated measuresANOVAof average CEs per vignette as a function of type

of vignette (3 levels), experimental condition (3 levels), and the influence of theposition of the Linda vignette on responses to the other vignettes (Linda vignettefirst versus last) revealed a highly significant effect for type of vignette,F(2, 276)58.35,p , .001, and no other significant effect. Figure 1 presents percentage ofCEs as a function of type of vignette, with the latter arranged in order ofincreasing elicitation of representativeness responses (as indicated in Fig. 2). It

FIG. 1. Percentage conjunction errors as a function of vignette type (Ab.Nat., Abstract-Natural;Con.Nat., Concrete-Natural; Con.Unn., Concrete-Unnatural). The stimuli are arranged in order ofincreasing unnaturalness, as determined by their rate of eliciting representativeness reasoning, asindicated in Fig. 2.



can be seen in Fig. 1 that the rate of CEs does not vary directly with therepresentativeness-eliciting properties of the vignettes. Rather, the abstract-natural vignette, which elicited a relatively low rate of representativeness reason-ing (see Fig. 2), elicits a relatively high rate of CEs. Despite attempts to reasonstatistically, about half of the participants made CEs in their responses to thisvignette because they apparently did not know the conjunction rule. Conjunctionerrors to the remaining vignettes do arrange themselves in the order of therepresentativeness-eliciting properties of the vignettes, with the concrete-naturalvignettes producing relatively few CEs and the concrete-unnatural vignettesproducing considerably more CEs, as anticipated. The Linda vignette, whichelicited the most representativeness reasoning, also elicited the most CEs.In order to determine the effect of treating right responses for wrong (represen-

tativeness) reasons as wrong responses, two parallel ANOVAs (excluding theLinda vignette) were conducted of CEs in the Reasons condition similar to the onereported for the total group. In one of the ANOVAs, the dependent variableconsisted of wrong responses (CEs), ignoring reasons; in the other, right re-sponses for wrong reasons were treated as wrong responses. In the analysis that

FIG. 2. Percentage representativeness reasoning as a function of vignette type (Ab.Nat., Abstract-Natural; Con.Nat., Concrete-Natural; Con.Unn., Concrete-Unnatural).

13LINDA PROBLEM


did not consider reasons, the vignette effect did not approach significance,whereas in the analysis that took reasons into account, the vignette effect wasmarginally significant (p5 .08). When similar analyses were conducted thatincluded the Linda vignette in the concrete-unnatural category, again, the standardprocedure did not produce a significant difference for vignette type. However, theprocedure that took reasons into account produced a highly significant effect forvignette type (p, .0001).In conclusion, as expected, the concrete-natural problems produced a relatively

low rate of CEs, and the abstract-natural and concrete-unnatural problemsproduced considerably higher rates of CEs. In addition, as further anticipated,treating right responses for wrong reasons as wrong responses increased thestrength of the relations.Reasons given and principles endorsed as dependent variables.AnANOVA in

which representativeness reasons in the Reasons condition was the dependentvariable revealed a highly significant effect for vignette type,F(2, 92)5 54.60,p, .001, and no other effect was significant. As can be seen in Fig. 2, the resultsare consistent with expectation. The order of the stimuli arranged by increasingrepresentativeness responses was as follows: abstract-natural (where statisticalreasoning was highly prevalent), concrete-natural, and concrete-unnatural. It canalso be seen in Fig. 2 that the Linda vignette elicited considerably morerepresentativeness reasoning (83%) than the other concrete-unnatural vignettes(66%). These findings provide a manipulation check on the construction of thegraded levels of the concrete-unnatural vignettes. It also provides a dimensionalong the abscissa of increasing representativeness properties of the stimuliagainst which the other variables can be examined (as in Figs. 1 and 3).A similar ANOVA of endorsement of incorrect principles in the Principles

condition also produced a significant effect only for vignette type,F(2, 92)5

3.14,p , .05. It can be seen in Fig. 3 that the relation of number of incorrectstatistical principles endorsed as a function of increasing representativenessproperties of the stimuli is V-shaped. Participants produced amoderately high rateof endorsement of incorrect rules for the abstract-natural vignette (36%), a lowerrate for the concrete-natural vignettes (25%), and a moderately high rate for theconcrete-unnatural vignettes (37%). The relatively high rate of endorsement ofwrong statistical principles for the abstract problem (see Fig. 2) indicates thatmany participants were unable to identify the conjunction rule when it waspresented. The highest rate of endorsement of incorrect statistical rules occurredfor the Linda problem (48%). Thus, with the exception of the abstract-naturalproblem, which elicited a high rate of statistical reasoning but a low rate ofendorsement of the correct statistical principle, the results for the Principlescondition replicate the results from the Reasons condition. Not surprisingly, therewas also a higher rate of endorsing correct principles for the concrete-natural andconcrete-unnatural vignettes in the Principles condition than of spontaneouslythinking of correct reasons in the Reasons condition.The significant effect for vignette type in the Principles condition is particularly



interesting because it indicates that, rather than endorsing the same principleacross vignettes, participants changed the principles they endorsed according tothe content of the vignettes. This was particularly evident in responses to theLinda vignette.

The Influence of Statistical Sophistication

Correlations were computed between CEs produced across all vignettes andboth measures of statistical sophistication: the ability to solve the quantitativeconjunction problem and exposure to a statistics course. The correlations werecontrolled for gender, order, and experimental group by partialling out theireffects. The correlation of CEs with having taken a statistics course did notapproach significance (r 5 2.09), whereas the correlation of CEs with the correctresponse to the quantitative conjunction problemwas highly significant (r 5 2.29,p, .01).In order to examine the effect of treating right responses for wrong reasons as

FIG. 3. Percentage incorrect principles endorsed as a function of vignette type (Ab.Nat., Abstract-Natural; Con.Nat., Concrete-Natural; Con.Unn., Concrete-Unnatural). The stimuli are arranged inorder of increasing unnaturalness, as determined by their rate of eliciting representativeness reasoning,as indicated in Fig. 2.

15LINDA PROBLEM


wrong responses, the correlations for the Reasons condition (the only group withthe requisite data) were recalculated, treating right responses for wrong reasons aswrong.With order and gender controlled, the correlation of CEs with the ability tosolve the statistics problem increased to2.63 (p, .001), whereas with havingtaken a statistics course, it remained nonsignificant.

DISCUSSION

Can the High Rate of Conjunction Errors to the Linda Problem Be Attributed tothe Violation of Conversational Rules Rather Than To an Experiential Modeof Information Processing?

As expected, completely disclosing information reduced CEs to the Lindaproblem over what was obtained in previous similar studies. However, despiteinstructions to regard all problems as essentially statistical and to ignore extrane-ous information, the reduction was far from complete. The majority of partici-pants continued to make CEs, and over 80% in the Reasons group reported thatthey engaged in representativeness, not statistical, reasoning. There are twoimportant conclusions that follow from these observations. One is that the essenceof the difficulty of the notorious Linda problem cannot be attributed to theviolation of implicit conversational rules, but is consistent with an explanation interms of the principles of experiential processing, as proposed by CEST. The tworelevant principles are that the experiential system is a ‘‘natural,’’ heuristic systemthat evolved to deal automatically and efficiently with everyday events, and that itencodes events in the form of concrete representations. From this, the importanceof a simultaneous classification of conjunction problems according to twodimensions, natural–unnatural and concrete–abstract, was derived. The results ofthe present study provide additional support for the importance of the intersectionof these dimensions beyond the limited support previously obtained (Epstein,Denes-Raj, & Pacini, 1995), which, of course, contributes to the construct validityof CEST.The other, not unrelated, conclusion is that processing in the experiential mode

can be sufficiently compelling in certain situations as to override processing in therational mode, even when the latter is equally accessible. Thus, many people whowere able to identify the correct rule for solving the Linda problem elected not toapply it. Moreover, their selection of principles varied according to the content ofthe vignettes, rather than being applied consistently across situations. Here, again,we find that although participants ‘‘know better,’’ they often prefer to behaveaccording to their experiential processing.These findings extend the generality of a similar conclusion about the compel-

ling quality of heuristic processing from our research in another domain: theratio-bias phenomenon (Denes-Raj & Epstein, 1994; Epstein, 1994; Kirkpatrick& Epstein, 1992). In that research, participants acted against their better judgmentby selecting a less favorable probability over one they recognized was morefavorable from an objective, rational perspective. The intuitive appeal of the less



favorable probability could be explained by the operating principles of theexperiential system, particularly the concretive principle.Other evidence that people prefer to process information intuitively rather than

rationally in some situations, even when instructed to behave otherwise, isprovided by research on lay people’s hypothesis testing. It was found that mostparticipants, despite being told to apply scientific principles of hypothesis testing,including a consideration of disconfirming evidence, attended to confirmingevidence and ignored disconfirming evidence (Gorman & Gorman, 1984; Gor-man, 1986). Such behavior is consistent with the concretive and affirmative-representation principles in CEST, according to which the experiential system canrepresent more readily positive than negative instances of behavior (Pacini &Epstein, 1996).In support of prediction, providing the conjunction rule to participants in the

Principles condition had little effect on CEs beyond the effect produced by thedisclosing instructions. This can be explained by the assumption that there wasnothing further that could be usefully directed toward the rational system once thecompletely disclosing information had been provided. On the other hand, and insupport of prediction derived from the operating principles of the experientialsystem, providingexperiencewith concrete-natural problems designed to activatelatent knowledge in the experiential system did have a significant additionaleffect. As this effect occurred in the absence of providing new information, and asthe reduction in CEs occurred in the absence of an increase in endorsements of thecorrect principle, it is consistent with the interpretation that the influence of theconcrete-natural problems operated not through promoting explicit awareness ofthe conjunction rule, but by activating latent intuitive knowledge.

Methodological Considerations

Consistent with previous research (Epstein, Denes-Raj, & Pacini, 1995), theimportance of obtaining information about reasons for responses was clearlyindicated. As in the previous research, a substantial proportion of participantsobtained right answers (avoidance of CEs) for ‘‘wrong’’ (representativeness)reasons. Since the reasoning in such cases is essentially no different from thereasoning that is the major cause of CEs, it is more appropriate to treat rightresponses for wrong reasons as wrong, rather than as correct, responses. Whenthis was done, the magnitude and coherence of relations with other variables, suchas with a measure of statistical sophistication, was considerably increased, aspredicted.An additional reason for unconfounding correct responses arrived at by correct

and incorrect reasoning is that in the absence of such correction the degree ofrepresentativeness reasoning is likely to be seriously underestimated and, relat-edly, the underlying process improperly understood. As high as the rate of CEs tothe Linda problem has been reported to be, the true rate of representativenessreasoning is even greater than has up to now been inferred from CEs. Moreover,this is particularly true under certain conditions, such as when the constituents of a

17LINDA PROBLEM


conjunction are both unlikely. In the absence of the recognition that correctresponses under such circumstances are often spurious, the results are apt to bemisinterpreted as indicating improved reasoning for a particular condition.One way to unconfound CEs and type of reasoning is to have participants

report the reasons for their responses, as in the Reasons group in the present study.This can be accomplished in the study itself, or by preliminary testing with adifferent group. An alternative procedure is to use vignettes that rarely elicit rightanswers for wrong reasons. For example, if the constituents in the Linda problemconsisted of two compatible constituents, such as Linda being a feminist andpro-choice, then people who engaged in representativeness reasoning could beexpected to almost invariably rate the probability of the conjunction as greaterthan that of its constituents. Almost all such participants would make CEs, andthere would be very few, if any, right answers for wrong reasons. Of course, giventhe surprising results that have been obtained repeatedly with the Linda problem,no such assumption should be accepted without having been confirmed byempirical demonstration.A caveat is in order about obtaining information on underlying thought

processes by providing selections from a list of correct and incorrect principles, asin our Principles group, and as has been practiced by others (e.g., Dulany &Hilton, 1991). Although such a procedure is useful for certain purposes, such asexamining the influence of providing the correct principle, it is not useful, as ourdata well demonstrate, for examining the reasoning that spontaneously occurs inresponse to conjunction problems, and is the principle source of CEs.A final methodological lesson from our results is that the inference of statistical

sophistication from exposure to statistics courses, which has been the usualprocedure, is seriously deficient. As we have demonstrated elsewhere (Epstein,Denes-Raj, & Pacini, 1995) and confirmed here, a more defensible measure ofsophistication about the conjunction rule, one that produces stronger and moreconsistent findings across studies, is to conduct a direct statistical test of the rulein the form of simple quantitative problems. This is hardly surprising consideringthat there is no guarantee that individuals who have taken a statistics course wereexposed to, learned, and retained explicit knowledge of the conjunction rule. Yet,self-evident as it may appear, it warrants emphasis, as it has been routinelyignored.

APPENDIX

Vignette TranscriptsVignette AVB.Assume that the likelihood of A occurring is very high and the

likelihood of B occurring is very low.Check which of the following you believe islesslikely.____ B will occur.____ A and B will both occur.

Vignette LOT.John buys two lottery tickets. One ticket is from the Pennsylva-nia state lottery and has a very low chance of winning. The other ticket is from a



local Cub Scout raffle and has a relatively high probability of winning.When Johnbought the tickets he was in a terribly confused state because he had an awfultoothache. In fact, he thought the tickets were for a movie. However, when hetried to turn them in at the movies, they refused to accept them. Nevertheless, heheld on to the tickets.Check which of the following you believe islesslikely.____ The Pennsylvania state lottery ticket will win.____ The Pennsylvania state lottery ticket and the Cub Scout raffle ticket

will both win.Vignette CAR.John goes to the auto races. He bets on two races. In one race, he

bets on a car driven by an experienced driver with a long history of winning races.In a different race, he bets on a car driven by an inexperienced driver who has wonvery few races.Check which of the following you believe islesslikely.____ The car driven by the inexperienced driver wins.____ The car driven by the inexperienced driver and the car being driven by

the experienced driver both win.Vignette DOS.Dawn of Spring is a neurotic race horse who runs very

inconsistently. He is alert and eager when he runs in races where there are small,quiet crowds, but he gets very nervous and uncoordinated when he runs in races inwhich the crowds are large and noisy. He is first entered in a noisy race in NewJersey and one week later in a quiet race in Delaware.Check which of the following you believe islesslikely.____ Dawn of Spring will win the noisy race.____ Dawn of Spring will win both the noisy race and the quiet race.

Vignette MARY.Mary is 28 years old. She is intelligent, friendly and outgoing.She studied physical therapy in college, and was a member of the volleyball team.She is very tall, athletic, and competitive.Check which of the following you believe islesslikely.____ Mary has green eyes.____ Mary has green eyes and plays basketball.

Vignette PAUL.Paul is 34 years old. In high school he was poor in mathematicsand good in music.Check which of the following you believe islesslikely.____ Paul is an accountant.____ Paul is both an accountant and plays jazz for a hobby.

Vignette LIN.Linda is 31 years old, single, outspoken, and very bright. Shemajored in philosophy in college. As a student, she was deeply concerned withissues of discrimination and social justice, and also participated in anti-nucleardemonstrations.Check which of the following you believe islesslikely.____ Linda is a bank teller.____ Linda is both a bank teller and a feminist.

19LINDA PROBLEM


REFERENCESCosmides, L., & Tooby, J. (1996). Are humans good intuitive statisticians after all? Rethinking some

conclusions from the literature on judgment and uncertainty.Cognition,58,1–73.Denes-Raj, V., & Epstein, S. (1994). Conflict between experiential and rational processing: When

people behave against their better judgment.Journal of Personality and Social Psychology,66,819–829.

Dulany, D. E., & Hilton, D. J. (1991). Conversational implicature, conscious representation, and theconjunction fallacy.Social Cognition,9,85–110.

Epstein, S. (1994). Integration of the cognitive and the psychodynamic unconscious.AmericanPsychologist,49,709–724.

Epstein, S., Denes-Raj, V., & Pacini, R. (1995). The Linda problem revisited from the perspective ofcognitive-experiential self-theory.Personality and Social Psychology Bulletin,11,1124–1138.

Epstein, S., & Donovan, S. (1995). [Is the Linda problem a true conjunction problem? Implicationsfrom frequency versus probability versions and individual differences in analytical style.]Unpublished data.

Epstein, S., & Pacini, R. (1995). [The Linda problem: A true conjunction problem, or does theconjunction rule not apply to multiple attributes in a single person?] Unpublished data.

Epstein, S., Pacini, R., Denes-Raj, V., & Heier, H. (1996).Individual differences in intuitive-experiential and analytical-rational thinking style. Journal of Personality and Social Psychology,71,390–405.

Fiedler, K. (1988). The dependence of the conjunction fallacy on subtle linguistic factors.Psychologi-cal Research,50,125–129.

Gigerenzer, G. (1991). How to make cognitive illusions disappear: Beyond heuristics and biases.European Review of Social Psychology,2,83–115.

Gorman, M. E. (1986). How possibility affects falsification on a task that models scientific problemsolving.British Journal of Psychology,77,85–96.

Gorman, M. E., & Gorman, M. E. (1984). A comparison of disconfirmatory, confirmatory, and acontrol strategy on Wason’s 2, 4, 6 task.Quarterly Journal of Experimental Psychology, A36,629–648.

Grice, H. P. (1975). Logic and conversation. In D. Davidson & G. Harman (Eds.),The logic ofgrammar.Encino: Dickenson.

Kirkpatrick, L. A., & Epstein, S. (1992). Cognitive-experiential self-theory and subjective probability:Further evidence for two conceptual systems.Journal of Personality and Social Psychology,63,534–544.

Lumney, G. H. (1970). Using analysis of variance with a dichotomous variable: An empirical study.Journal of Educational Measurement,7,263–269.

Pacini, R., & Epstein, S. (1996).Lessons on intuitive processing from the ratio-bias phenomenon:Principles of operation and interaction with rational reasoning.Manuscript submitted forpublication.

Schwarz, N. (1994). Judgment in a social context: Biases, shortcomings, and the logic of conversation.Advances in Experimental Social Psychology,26,123–162.

Sperber, D., & Wilson, D. (1986).Relevance: Communication and cognition.Cambridge, MA:Harvard Univ. Press.

Tversky, A., & Kahneman, D. (1983). Extensional versus intuitive reasoning: The conjunction fallacyin probability judgment.Psychological Review,90,293–315.

Yates, J. F., & Carlson, B. W. (1986). Conjunction errors: Evidence for multiple judgment procedures,including ‘‘signed summation.’’Organizational Behavior and Human Decision Processes,37,230–253.



the difficulty of the linda conjunction problem can be attributed to its simultaneous concrete and...

Documents