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Repeated reasoning on logic table problems.

Many institutions of higher education tout their graduating students' ability to engage in "critical thinking" as being among the most important talents of the educated person (Lloyd & Bahr, 2010). Critical thinking is a complex phenomenon of related cognitive acts, one of which includes the ability to carry out advanced deductive reasoning. Given the importance of critical thinking, it is therefore somewhat surprising that curriculum designers and other planners seem to know relatively little about the cognitive science of human reasoning, including the barriers that seem to stand in the way of successful reasoning.

Among those barriers to successful deductive reasoning, for example, there are well-known belief bias effects. Evans, Barston and Pollard (1983) showed that when there was a conflict between the believability of a logical argument and validity considerations, the believability criterion overrode the logical considerations, which in turn strongly suggests that belief-bias is based on the believability of a logical argument's conclusion (Evans & Over, 1996). In addition, people perform rather poorly on logical tasks in which they are asked to identify logical conclusions (Evans, 2002), generally performing at levels far below 100% correct. However, people nevertheless recognize logical arguments at rates that are well above chance (Evans, 2002). Moreover, they are sensitive to the effects of instructions to "think logically" (Evans, 2000). People thus adapt their reasoning processes to an extent when instructed to base conclusions on logical necessity rather than belief. That is, the instructional effect can ameliorate the belief-bias effect (Evans et al., 1994; George, 1995; Stevenson & Over, 1995). Furthermore, processing time and speed of reasoning are known to influence reasoning accuracy (Evans & Curtis-Holmes, 2005; Forgues & Markovits, 2010). Evans and Curtis-Holmes showed that when reasoners were forced to respond within 10 seconds on a syllogistic reasoning task, the number of logically correct responses declined and the influence of belief bias increased. Finally, Sloman (1996, 2002) has found that in evaluating argument strength, people are sometimes aware that an initial, categorically-derived response may lead to a judgment of lower argument strength than will a later, logically-necessary, response. Sometimes these reasoners have suggested that, although their two responses contradicted each other, both responses still seem sensible to them.

These and other phenomena of human reasoning are often explained within a theoretical context known as the dual-systems (sometimes, dual-process) approach (Evans, 1984; 1989; Evans & Over, 1996; Sloman, 1996, 2002; Stanovich & West, 2000, 2002). According to this view, each of the two systems can be understood as a self-contained faculty for representing and acting upon the world (Evans, 2009). The first system, also frequently called the heuristic system, reasons on the basis of associations apparently formed by covariation, probability, or temporal proximity of stimuli. The reasoning steps taken by the heuristic system are not available to consciousness, only the result is, and this result is generally quickly available. Because it is experientially-based, the heuristic system is context-bound, and thus always potentially biased. This system's characteristics can be contrasted with a second, and perhaps weaker, system called the analytic system. Unlike the heuristic system, the analytic system places attentional and memory demands on the overall system. Reasoning is accomplished in a series of steps whose processing is consciously available, and is usually deliberate and slow. However, because the material for reasoning may be represented more abstractly than that used by the heuristic system (Rips, 1994), the analytic system is more flexible, and less context-bound than the heuristic system is, and it may be able to address and manipulate the concepts of logical necessity and validity.

Although there has been a great deal of empirical evidence amassed in support of the dual-systems position (see Evans, 2008), this theoretical explanation has nevertheless been questioned relentlessly and persistently on methodological, conceptual, and empirical bases (e.g., Gigerenzer & Regier, 1996; Hammond, 1996; Keren & Schul, 2009; Osman, 2004).

Osman (2004) proposed an alternative to the dual-systems account based on an adaptation of work by Cleeremans and colleagues (Cleeremans & Jimenez, 2002; Gaillard, Destrebecqz, Michiels, & Cleeremans, 2009) in the domain of implicit learning. In their model, known as Dynamic Graded Continuum (DGC) theory, Cleeremans and Jimenez (2002) argue that the representation that is used in learning can be placed on a three-dimensional continuum involving the representation's strength, distinctiveness and stability. They use a neural-network paradigm to describe these dimensions. In their model, the cognitive system is organized as a set of hierarchically arranged, interconnected processing modules, which create and process the representation preconsciously, although the representation may become conscious as its quality improves by advancing on the three dimensions. In their connectionist model a representation's strength is defined as the amount and level of activation of the processing units in the neural network. In learning tasks, active processing of the material increases the strength of the associations among the processing units. Increases in the strength of a representation are seen in recall and transfer measures which indicate the extent to which the representation is strong enough to be reliably called up by a central executive. Stability refers to the length of time that the representation remains active during processing. When materials are processed briefly, subjects are not necessarily accurate or confident in their judgments that the items have been presented, but such measures increase a function of the materials' display time. The distinctiveness dimension corresponds to the representation's discriminability from other superficially similar (feature-based) representations. As the representation advances in discriminability, there is an expectation of greater accuracy and confidence in part-to-whole judgments.

In her use of the DGC perspective as an alternative to the dual-systems position, Osman (2004) focused chiefly on a couple of characteristics of the dual-systems perspective: The speed of processing issue and the awareness issue. In her view, the facts that some reasoning can seemingly be done quickly, and without awareness of processing, while other reasoning is done more deliberately and with awareness, does not necessarily imply that two systems are at work. She suggested an application of the Cleeremans and Jimenez (2002) account as an alternative explanation. As in the Cleeremans and Jimenez (2002) theory, reasoning in a particular domain may begin with an implicit phase. Implicit reasoning involves making a set of abstractions or inferences without the concomitant awareness of them. The abstractions or inferences occur unintentionally, are not susceptible to conscious control and are therefore not directly accessible to manipulation (although they may still be capable of influencing explicit processes). As the representation for reasoning gains in strength, distinctiveness, and stability through the reasoner's repeated exposure to it, the representations formed become more coherent and distinct, and the reasoned may have greater awareness that he or she did previously. The DGC position defines this stage of explicit reasoning as having awareness of the abstractions or inferences that are made, which can be expressed as declarative knowledge. The abstractions or inferences are available to conscious control and this allows the inferences to be modified directly because they are accessible. There is conscious control of the representations because one has metaknowledge of them. The representations have a high rate of activation, and can be reliably recalled from memory because they are stable enough to become registered in working memory. This form or stage of reasoning corresponds most closely to the operation of the analytic system in the two-systems perspective. Finally, with additional exposure to the representation that is being used for reasoning, the reasoner may show signs of automatic reasoning, which corresponds to the operation of the heuristic system. Corresponding to the learning theory of Cleeremans and Jimenez (2002), automatic reasoning is skill based and deliberately acquired through frequent and consistent activation of relevant information that becomes highly familiarized. This type of reasoning enables abstractions or inferences to be made without any control because the representations are enduring, well-defined, and stable through repeated use. Thus a skilled reasoner or logician will automatically recognize which premises are relevant, what the correct set of inferences are that follow, and what conclusion should be drawn. This chain of inferences will, if the task is highly familiar, be executed automatically. For example, Galotti, Baron, and Sabini (1986) compared poor, good, and expert syllogistic reasoners and found that experts responded faster than either of the other groups and displayed accurate metaknowledge of their reasoning behavior. In contrast to this situation, Osman (2004) suggests that implicit (or heuristic) reasoning is likely to result from situations in which the reasoners are unfamiliar with the reasoning domain. In that case, the chains of inference that lead to a conclusion are not retraceable, or verbalizable, because they are new and unfamiliar, and as such, are not associated with the appropriate metaknowledge.

In summary, Osman (2004) has presented a cogent alternative to the dual-systems perspective that warrants attention. However, despite the clarity, credibility, and obvious relevance of Osman's alternative perspective, there have not apparently been any systematic attempts to address Osman's criticisms of the dual-systems approach directly.

One of the key issues for the alternative DGC account of reasoning concerns the organization and structure of the representation that is used for reasoning. The rate at which inferences are reached, their accuracy, and perhaps the awareness of the underlying process, are all a function of the quality of the representation underlying the reasoning process. The DGC position suggests that people create and develop a representation for dealing with a specific reasoning problem, or class of such reasoning problems. As the reasoner works with that representation, each instance of processing serves to increase the strength, distinctiveness and stability of the representation. If the level of this representation is sufficiently abstract, then the presentation of an apparently new problem that shares the same structure should be processed more quickly than would a reasoning problem that did not share the same underling structural features. To the extent that the reasoner has deductive competence, there should be no loss in deductive accuracy. However, if the reasoner is presented with logical problems that demand different representations to be created to solve them, then the reasoner may be "stuck" at the deliberate, analytic level of processing. Under those conditions, there shouldn't be any speed-up in the rate of inference-making across a series of such problems. The DGC position therefore predicts that at least some aspects of the reasoner's performance, including the reasoner's speed and accuracy in deduction, should be susceptible to manipulations that influence the quality of the representation being used as a basis for reasoning. On the other hand, the dual-systems position predicts a different pattern of responses in repeated reasoning episodes. To the extent that the reasoner must use his or her analytic system, there shouldn't be any speed-up in performance across a series of reasoning problems, regardless of whether they share any structural identities. Moreover, to the extent that the reasoner can be cued to "be logical," and to the extent to which he or she has deductive competence, then deductive accuracy should suffer if a speed-up in performance occurs.

The current paper reports the results of three experiments, two of which involved attempts to manipulate the structure of the representation used for reasoning. In Experiment 2 the structure of the representation was varied by creating so-called "logic table" problems in which different variables were "connected" to each other in the problem's text. If the structure of the problem's representation is specific enough to include representations among only those variables which were connected in the problem, then presenting problems with different connections among the variables should defeat the improvement in the representation that DGC theory demands in order to see a speed-up in performance. In Experiment 3 the structure was varied by creating logic table problems whose solution paths were different, in terms of the inferences that could be made from the information that was given, given the deductions that had become available to be made up until that point in the problem. If the structure of the problem's representation is specific enough to include the placement values in the inference chain, then this variation in structure should defeat the improvement in the representation that DGC theory demands in order to see a speed-up in performance.

The three experiments reported here used basically the same procedure. Participants were run individually. After being greeted by the experimenter, each participant was told that the experiment was to be conducted in two phases. The experimenter first described a set of 16 conditional reasoning problems whose data were used in another series of studies, and are not included here. While the participants worked on these problems at their own pace, the experimenter left the lab for 10 minutes. After the participants completed the conditional reasoning problems, they were invited to complete a set of three (or in the first experiment, four) logic table problems whose details are described in more depth in the materials section of each study. The experimenter told the participants that they could take as much or as little time as they liked on each problem, and that they could write whatever they wanted on the problem sheet. The participants were also informed that once they had completed a specific problem, they were not to go back to an earlier problem.

EXPERIMENT 1--REPEATED REASONING ON LOGIC TABLE ISOMORPHS

The Dynamic-Graded Continuum (DGC) position has been offered as an alternative framework for understanding some of the findings obtained by the proponents of the dual-systems view. According to DGC, reasoning might be accomplished by a system that reasons from a single structural representation whose quality is based on changes in its strength, distinctiveness, and stability--dynamic properties that are a function of the structure's repeated processing and use. In the literature on problem solving, problems that share an underlying structural representation, and differ from each other only in their superficial details, or "cover story" are referred to as "isomorphs" (Newell & Simon, 1972). In mathematics, examples of isomorphic problems would be solving a series of problems in algebra that require constructing and solving a set of equations whose variables had the same relationship in each problem. Although very little is known about what happens when people engage in repeated episodes of reasoning across a set of problems that share a clear structural relationship (that is, problems that are isomorphic to each other), the DGC framework suggests that people will speed up in their solution of such problems that share a structural relationship, because the representation of their structure is being strengthened by repetition. Dual-systems theory predicts that people can be cued to use their analytic or logical systems by certain featural aspects of the problem, thus making them somewhat immune to whatever effects of shared structural resemblance there may be. For example, if people are asked to solve logic problems that are introduced as "logical" and that are accompanied by an answer sheet in the form of a "grid" or matrix that invites a stepby-step processing and recording of specific inferences, then such individuals may stay in an analytic mode of processing, or stay in it longer, than people solving problems that are not presented that way. Consequently, the effect of the repetitive structure of the logic problems, even if it is observed, may be attenuated for the participants who are supplied with that kind of detailed answer grid. However, if this attenuation in speed is observed, we may also expect that participants who have been cued to use their analytic systems (by the presence of the detailed answer grid grid) will solve logic problems with greater accuracy than will participants who are not cued.

Method

Participants The participants were 25 volunteers (16 women, nine men) from a Cognitive Psychology class who received an incentive of extra credit points totaling approximately 1% of the total points available in the course.

Materials A set of "logic table" problems was constructed as follows. First a "base problem" was obtained from a commercial subscription site (www.edHelper.com) that produces such problems for use by teachers in high school mathematics classes. The base problem consisted of two principal elements. There was a brief cover story describing some type of routine activity (e.g., going shopping, choosing a major, having breakfast together, etc.) of five named hypothetical students. The cover story also established five specific characteristics or objects, such as five different stores that one might shop in, five different majors that one might choose, and so on. The second element was a set of seven factual premises that established some relationship between one or two of the students and one or two of the attributes. For example, if the cover story was about choosing a major, the participants might read the statement "Neither Keisha nor Joe chose Psychology as a major." From this base problem, three isomorphic problems were created by changing the names and objects in the cover story. Only the minimum necessary changes in wording were made to the seven premises to make them consistent with the cover story in each of the problem's isomorphs. Each logic table problem appeared by itself on an 8.5 x 11 in. (21 x 27.5 cm) sheet of paper.

Design A between-subjects variable was created by adding a 5 x 5 matrix or "logic grid" to the problem, showing the names of each of the hypothetical students on the rows and the objects or attributes in the columns, with 25 blank lines in the body of the matrix. The participants who were randomly assigned to the "grid-present" condition saw this 5 X 5 matrix and were told that it could be "helpful" in reasoning about the problems. Moreover, they were instructed to write "Certain" "Possible" or "Impossible" on each of the blank lines as they drew conclusions about the connection between that student and the object. Participants randomly assigned to the "grid-absent" condition saw only the space for their answers below the problem's text.

Procedure In addition to the information in the "Overview of procedure" section, the experimenter explained that the solver's task was to deduce the link between each of the five students and the object or attribute, such as for example, what major had been chosen by each student. The experimenter did not explain to the participants how the grid may have been used in a strategic sense (for example, once the reasoner is "Certain" about a particular combination, then every other line in that row and column can be filled in as "Impossible").

Results

The first hypothesis is that participants in the grid-absent condition should show a speed-up in reasoning across the four isomorphs as their representation of the problem's structure increases in strength, distinctiveness, and stability (Osman, 2004). To test this hypothesis, a one-between (logic grid-present vs. logic grid-absent) X one-within (four reasoning isomorphs) ANOVA was carried out on response times to the four problem isomorphs. This hypothesis was supported. Participants solved their second and subsequent logic table problems in less time than that required for their first logic table problem, F(3, 69) = 23.92, p < .001, [[eta].sup.2.sub.p] = 58. Figure 1 shows the speed-up in logic table performance for both the grid-present and grid-absent participants. As it shows, there was a substantial reduction across the set of logic table problems with an overall mean of 399s for the first logic table problem, and a mean of 206 s for the fourth and final logic table problem. However, most of the overall effect was accounted for by the reduction in time from the first to the second problem.

The study's second hypothesis was not supported. There was no main effect on response time associated with the grid's presence or absence, F(1, 23) = .083, p = .776, [[eta].sup.2.sub.p] = .004: Overall, participants in the grid-present condition required no more time to solve the logic table problems (M = 267 s, SE = 23.4 s) than did participants in the grid-absent condition (M = 277 s, SE = 22.5 s). Similarly, there was no interaction of grid-presence or grid-absence with response time, F(3, 69) = 1.12, p = .346, [[eta].sup.2.sub.p] = 09. Basically, participants in the grid-present condition showed virtually the identical speed-up in solution time as that shown by the participants in the grid-absent condition.

Finally, there was an expectation about performance: Participants who may have been cued to use their analytic systems (by the presence of the logic grid) would indicate more correct answers on the logic table problems compared to the grid-absent participants who may not have been cued to use their analytic systems. However, performance on the logic table problems was virtually at ceiling for all four isomorphs in both the grid present and grid absent groups, thus nullifying the opportunity to do this analysis. There were very few wrong answers on the logic table problems by any of the participants.

Discussion

As the DGC alternative suggests, people did indeed show a speed-up in performance across a set of four isomorphic logic table problems. On their fourth logic table problem, people required only about 50% of the time they required to do their initial problem, with no fall-off in accuracy of deduction. Moreover the fact that the speed up was the same for grid-present and grid-absent participants also supports the DGC approach. It is interesting that the grid-present participants did take longer, although not significantly so, than did the grid-absent participants on their initial problem, suggesting that the grid-present participants were trying to "be logical" by using the grid. The presence of the grid was theorized to work as a cue prompting people to use their more deliberate analytic systems to solve the logic table problems. The expected result of this manipulation was that people in the grid-present condition would not show a reduction in logic table solution times, or at least, would show a smaller reduction than that seen for the grid-absent participants, a result that would support the dual-systems viewpoint. But the results did not turn out in that direction. It might be the case that the failure to observe this difference was because the participants in the grid-present condition did not take advantage of the grid to help solve the logic table problems. But this alternative explanation can be ruled out: Almost 100% of the participants in the grid-present condition filled in at least some of the values on each grid of each problem. Somewhat surprising was the fact that almost 100% of the participants in the grid-absent condition made up some form of organizational grid on their own, and doing so did not appear to slow them down. Finally, the expected result that would have provided the strongest support for the dual-systems perspective, namely that people in the grid-present condition were more accurate in their deductions than were people in the grid-absent condition, could not be tested. The modal response for each participant was to correctly deduce every combination on each problem, strongly suggesting that the logic table problems in this experiment were probably too easy to put this hypothesis to a fair test.

EXPERIMENT 2--CHANGING THE PATTERN OF CONNECTIONS AMONG VARIABLES

Although there was a speed-up in solution time across the set of problems used in Experiment 1, there was no structural variation in any of the logic table problems, and the DGC position predicts that the changes in the quality of the underlying representation used for reasoning occur only when the same structure is presented repeatedly. Also, the logic table problems used in Experiment 1 turned out to be too easy, with performance virtually at ceiling. Experiment 2 represents an attempt to present equivalent, but different, logic table structures to reasoners. Experiment 2 also features more complex problems than those used in Experiment 1. In the context of a more challenging logic table problem, it may be the case that the presence of a "logic table grid" induces the reasoner to use his or her analytic system to reason, thus reducing the effect of the speed-up in solution time seen in Experiment 1.

Method

Participants The participants were 48 volunteers (16 men, 32 women) from upper-division Psychology courses in Cognitive Psychology or Learning, in exchange for a small amount of extra credit, generally amounting to 1% of the total number of points available in the course.

Materials Logic table problems. Each problem was based on an example obtained from a commercial subscription site (www.ed Helper.com) that produces logic table problems for use by teachers in high school mathematics classes. Each problem consisted of the following elements. The first element was a brief cover story that identified four hypothetical students by first name, and described two kinds of everyday characteristics about them as a group. One characteristic was categorical (e.g., it might be the residence hall in which a student might live, or his or her major); the second characteristic had the property of magnitude (e.g., it might be the number of hours each student had earned toward graduation, or the starting salary of the student's first job). The second element, which I called the problem's "text," was a set of eight factual premises, each of which established an "is not a" relationship between a value on one of the characteristics and one or two values on a second characteristic. For example, a premise might assert that "The person who has 28 hours (i.e., a value of the scalar characteristic in this case) does not live in Greek Court or McKinney" (values of the categorical characteristic). Only the minimum necessary changes in wording were made to the eight premises to make them consistent with the cover story in each version of the problem. A space for the participant's answers was the next element of each logic table problem. In the answer space were three columns labeled with that problem's characteristics (i.e., the person's name, the categorical characteristics, and the magnitude or "scalar" characteristic). For each problem's answer element, the four values of one characteristic were explicitly stated in one column; there were blank lines for the values of the other two characteristics. The final element appeared only on the problems of the subjects assigned to the "logic Grid Present" condition. Two 4 X 4 matrices were printed below each problem. One matrix showed the values of two of the characteristics mentioned above (for example, the names of the students and the names of the residence halls) in the rows and columns, with 16 blank lines in the body of the matrix. The second matrix repeated the values of the characteristics in the columns, with the rows showing the values of the remaining characteristic, and 16 blank lines in the body of matrix. The characteristic shown in the columns was always the predicate of the "is not a" factual premises. In the example shown above, the characteristic shown in the columns would be the four values of the residence halls used (i.e., the four names of the residence halls used in the problem). Each problem appeared by itself on an 8.5 x 11 in. (21 x 27.5 cm) sheet of paper. Appendix A shows an example of one of the logic table problems that was used.

Varying the structure of logic table problems. Each premise in the problem's text linked instances of different variables with an "is not an" connector in the following way:

   [instance of Variable X] "is not an" [instance of Variable Y]


Four of the eight premises in the problem linked one instance of Variable 1 to an instance in Variable 3, and the four remaining statements linked one value of Variable 2 to an instance of Variable 3. If the student's first name was being used as Variable 1, the residence hall was being used as Variable 2 and the number of hours earned was being used as Variable 3, then the following two statements show how the problem's text would be structured:

   Miranda doesn't live in Greek Court.
   The person who has earned 64 hours does not live in Ford Hall.


For each problem, the "is not a" connections between the first two variables (personal names and residence halls) was mentioned, and the "is not a" connections between Variable 3 (hours earned) and Variable 2 were explicitly stated in the text. But the connections between the first two variables (personal names and hours earned) themselves were never explicitly mentioned. So for the example above, the student would not see any explicit "is not a" premises linking the name of the student with a number of hours that he or she might have earned. Each characteristic described above (i.e., student's first name, categorical or scalar characteristics) was used an equal number of times as Variable 1, Variable 2 or Variable 3.

The variable whose specific values were shown on the answer blanks was called the "Display" variable. One of the reasoner's tasks in the problem was to make inferences from the display variable to the correct values of the other two variables. The text always linked values of the display variable to values of one and only one of the other variables. In the example above if the students' names were the display variable, then a student's name would be linked only to the values of one other variable (in this case, the categorical variable). This variable whose values were not displayed on the answer blanks, but which were nevertheless linked in the problem's text was called the "Nondisplay" variable. In this case, the categorical variable (residence halls) is the "Nondisplay" variable. Finally, the variable that was neither displayed, nor linked in text with the display variable is the "Nonconnected" variable. In this case, the scalar variable (number of hours earned) is the "Nonconnected" variable. For the logic table problem shown in Appendix A, the students' names are the Display variable; the various jobs that each student could have are the Nondisplay variable, while the salaries that each student earns are the values of Nonconnected variable.

Logic table problems having different structures were created by rotating each characteristic (personal name, categorical, or scalar) into each of the variable types (Display, Nondisplay, Nonconnected) to create six different structural types as shown in Table 1. For example, as in Appendix A, where the personal name characteristic is the display variable, the jobs the students obtain are the nondisplay variable, and the salary characteristic is the nonconnected variable, the reasoner might look at the list of names (which are displayed) and make deductions "forward" from the text about the jobs that each student obtained. Finally, the reasoner might also reason forward from the list of names to the salary information, even though the students' names and salary information are never directly linked in the text. However, this kind of reasoning is not so readily available when the categorical characteristic becomes the display variable, and the personal name is the characteristic that is no longer displayed.

Design Participants solved three logic table problems, each of which had a different cover story. Personal names representing hypothetical students were used as a characteristic in each cover story, but the specific nature of the categorical characteristic was altered in each problem (such as the residence halls the hypothetical students lived in, or the home towns the hypothetical students were from, and so on). Similarly, the scalar characteristic might refer to the number of credit hours a hypothetical student had earned, or the GPA of the hypothetical student, etc. The participants were randomly assigned to either a "Different Structures" condition, or an "Isomorphs" condition. The participants assigned to the different structures condition attempted three problems each of which featured a different characteristic (person, categorical, scalar) in the three variable roles (Display, Nondisplay, Nonconnected). Participants randomly assigned to the isomorphs condition saw the same characteristic in the same variable role for each of their problems. Participants were also randomly assigned to a "logic Grid Present" or "logic Grid Absent" condition, which was completely crossed with the different structures vs. isomorphs variable.

Procedure The participants were instructed to deduce the identity of each person, and their values on the categorical and scalar variables. The "answer sheet" displayed blank spaces for each of the four values for each variable. The values of one variable (the Display variable) were already filled out for the participant. The participant's task was to deduce the values of the other two variables that were associated with the values that were provided. The participants were told they could write on the paper. Participants who were assigned to the "Grid Present" conditions were told the matrix showing the all the characteristics that were used in the problem could be "helpful" in reasoning about the problems, and these participants were given instructions to write "Certain" "Possible" or "Impossible" on each of the blank lines as they drew conclusions about the connection between that student and the object. No strategic information per se concerning this grid was discussed.

Results

Solution Time Effects The DGC approach predicts that the problem's representation is strengthened by repetition. Consequently, participants who solved structurally similar Isomorphs should have their representation strengthened, in comparison to those participants who solved logic table problems with different patterns of connections among the variables. To test this hypothesis, a 2 (Grid Present vs. Grid Absent) X 2 (Isomorphs vs. Different Structures) between-subjects ANOVA was carried out on the solution time data, which was treated as a repeated measures variable.

Consistent with the predictions of the DGC approach, there was a dramatic speed-up in solution times for participants across the set of three problems. Collapsing across all four conditions, the participants solved their second and third logic table problems in substantially less time than they required for their initial logic table problem, F(2, 43) = 21.25, p < .001, [[eta].sup.2.sub.p] = .497 (both the linear trend, F(1, 44) = 40, p < .001, and the quadratic trend F(1, 44) = 19.65, p < .001, were significant). The participants required 473.8 sec to solve their initial problem; they required 298 sec to solve their final problem, a reduction of 37%. This pronounced speed-up in solution times is shown in Figure 2.

However, contrary to the expectations of DGC theory, there was no main effect of the Isomorphs vs. Different Structures variable on overall solution times F(1, 44) < 1, p > .05. Participants in the Different Structures conditions showed the same speed-up in solution times across the series of logic table problems that participants in the Isomorphs conditions showed, a result that does not support the predictions of DGC theory. Similarly, dual-systems theory predicts that participants who were in the Grid Present conditions, and who were prompted to be logical, should solve more deliberately than those who were in the Grid Absent condition. However, this effect predicted by dual-systems theory was not observed, F(1, 44) = 1.26, p > .05.

Accuracy Effects: Scoring the Logic Table Problems Each logic table problem has a three-letter code designating which characteristic (Person, Categorical, or Scalar) was playing the role of the Display variable, the Connected variable, or the Nonconnected variable. The participant scored one point for each correct deduction that he or she made from the first variable stated to the third variable stated. In the example problem [PSC], the participant scored one point for each occupation they correctly wrote on the same line as the person's name, thus making an inference from the Display variable to the Nondisplay variable (D [right arrow] ND), coded as Display inferences. The participant also received one point for each correct deduction that he or she made from the first variable to the second variable. In the example problem [PSC], the participant scored one point for each salary correctly written on the same line as the person's name, thus making an inference from the Display variable to the Nonconnected variable (D [right arrow] NC), coded as Nonconnected inferences. Finally the participants scored one point for each correct deduction that he or she made from the second variable stated to the third variable stated. Thus the maximum score that a participant could earn on any given logic table problem was 12, and the maximum score he or she could earn across the set of problems was 36. The scores within each category of deduction were not completely independent of each other, because if an error was made by a participant, it typically led to a transposition of a second mistake. Similarly, errors in reasoning are not independent of each other across categories either. For example, if a participant made deductive errors in the D [right arrow] ND category, he or she was then bound to make errors on either the D [right arrow] NC category on the NC [right arrow] ND category. However, theoretically any two categories of deductions are independent of each other, and as a class, the D [right arrow] NC (Nonconnected) deductions should be more difficult than the D [right arrow] ND (Displayed) deductions.

According to the dual-systems viewpoint, reasoners in the Grid Present condition were prompted to use their analytic systems, and thus they should score higher than reasoners in the Grid Absent condition for each of the three types of deductions available (Display, Nondisplay, and Nonconnected). This hypothesis was tested in a series of three 2 (Grid Present vs. Grid Absent) X 2 (Isomorphs vs Different Structures) between -subjects ANOVAs using the reasoner's score on the Display, Nondisplay, and Nonconnected inferences as the dependent variable in each ANOVA (collapsing these inferences across the participants three solution attempts). Contrary to the expectation offered by dual-systems theory, there was no main effect of the Grid Present on any of the three types of deductions, [F.sub.Display] (1, 44) < 1; [F.sub.Nondisplay] (1, 44) = 1.88, p = .18; [F.sub.Nonconnected] (1, 44) < 1. However, as in Experiment 1, the reasoners' composite deductive score across the set of three problems (M = 32.23) was still close to the maximum (36), a finding that will be discussed next.

Discussion

With regard to the solution time results, as predicted by the DGC position, but not by the dual-systems position, reasoners in the Isomorphs condition required substantially less time to solve their final logic problem than they did their initial problem, with no loss in deductive accuracy. Complicating the interpretation, however, is the fact that the reasoners in the Different Structures condition showed a virtually identical speed-up in solution time with no loss in deductive accuracy--a finding that is not necessarily predicted by either position. And although dual-systems theory predicts that reasoners who persistently use their analytic systems to solve these problems should be more deliberate, accurate, and slower in comparison to those reasoners who may default into some use of their heuristic systems, no such effects were observed. Actually, the participants in the Grid Present conditions, who technically should have been the ones to continue using their analytic systems to reason, solved the problems in marginally less time than did those in the Grid Absent conditions. But this slight difference may have been the result of the fact that almost every participant in the Grid Absent conditions created some form of logic grid on paper for themselves. This implies that the marginally faster solution times observed for the Grid Present participants were the result of not having to create an external memory.

The DGC alternative position makes a claim that reasoning is accomplished from a representation of the task whose strength, distinctiveness, and stability increase with repeated presentations (Cleeremans & Jimenez, 2002; Gaillard, Destrebecqz, Michiels, & Cleeremans, 2009; Osman, 2004). Thus, the DGC position can account for the speed-up in solution time, with no loss deductive accuracy, seen in the Isomorphs condition. What the DGC position cannot immediately explain is how the participants in the Different Structures condition were able to show the same effect. One possible alternative explanation is that the manipulation used in the problems to create different structures, namely the rotation of different variable categories into different roles in the problem's text, may not have been pronounced enough to require a different structural representation, an issue that raises some questions about the specificity and content of any representation that would be used for reasoning.

One of the claims of the dual-systems position is that an explicit or implicit prompt to "be logical" in their solution attempts will result in more deliberate and accurate deductive performance (Evans, 2000; 2002). But this claim is predicated on the difficulty of the logic task in the first place. If the task does not necessarily require much deductive competence, then the implicit prompt in this experiment (i.e., the presence of the Logic Grid) or the explicit instructions, may not have sufficed to induce the participants to use their analytic systems. Although the participants required significantly more time to solve these problems than did the participants in Experiment 1, they nevertheless were able to successfully make almost all of the deductions (mean performance = 89%). This suggests that the set of problems may not have been a particularly sensitive measure of deductive competence, and that particular claim of dual-systems theory may await a more conclusive test.

EXPERIMENT 3--DIFFERENT PATHWAYS THROUGH THE MATRIX

Experiment 2 showed that the structural manipulation of having different connections among the variables mentioned in the problem's text did not have any effect on the time required for successive solutions, or on the accuracy of the deductions made.

However, possibly countervailing the argument that the results do not support DGC theory, it may not be the case that the structural manipulation that was used in Experiment 2, that of altering the pattern of connectivity between the variables for participants in the Different Structures conditions, actually required a different structure for reasoning.

However, there may be other structural manipulations that do not involve the pattern of connections among the variables. For example, although it may not be immediately obvious that this is the case, it is true that the information initially presented in the problem's text affords some particular deductive inferences more readily than it affords others. That is, some deductions require fewer reasoning steps from the information presented, while others will often require a more lengthy "chain" of inferences, built partially on the deductions that are more immediately apparent. Thus it is possible to create Isomorphic logic table problems that are built around the same chain of inferences. Even though the problem's cover story may vary, in a structural sense each premise in the problem's text can be designed to convey the same information about some particular predicate value of some particular variable. When the logic table problem is designed that way, each statement in a given problem can be mapped directly onto a corresponding statement in a subsequent problem, even though the problem's superficial content will be different. And by creating a sequence of problems in that way, the reasoning "pathway" that the solver may take to arrive at particular deductions is similarly constrained across subsequent logic table problems. On the other hand, it is possible to create two (or more) different textual presentations of a logic table problem that share only the same answers. In that case, there can be no direct mapping of statements of one problem onto another. Under these circumstances, it seems that a functionally different representation will be created corresponding to the reasoning pathway that the solver will be led onto. This analysis is the underpinning for the structural manipulation used to create Isomorphs and Different Structures in Experiment 3. According to the DGC position, reasoners who can avail themselves of a representation that incorporates the same reasoning pathway should show the expected speed-up in solution times, accompanied by perhaps an improvement in deductive accuracy, but at least no loss in deductive accuracy. Thus there should be reduction in solution time for reasoners in the Isomorphs condition, but the not Different Structures condition. In addition to predicting better overall deductive performance for reasoners in the Isomorphs condition, the DGC position also makes an intriguing prediction about the sequence of inferences that reasoners undertake. If reasoners are able to build a representation around the reasoning pathway, then the DGC position suggests that reasoners solving Isomorphs should be able to make more "deep" inferences (i.e., inferences that can only be made late in the chain of inferences) as they solve subsequent logic table problems compared to reasoners who are solving problems that have a different reasoning pathway for each problem. According to the expectations of the DGC position, these reasoners may not be able to get as deep into the problem, in the same amount of time, as reasoners who are solving Isomorphs. Experiment 3 was designed as a test of these expectations.

Method

Participants There were 58 undergraduate and Masters-degree participants in the study. Forty-four participants volunteered from upper-division Psychology courses for extra credit amounting to 1% of the total number of points available in their courses. Fourteen participants, including all of the Masters-degree participants, volunteered for an incentive of $10 USD.

Materials

Logic table problems. As in Experiment 2, each participant was also asked to solve a set of three reasoning problems described as "critical reasoning" problems. Each of the problems had a different cover story, and dealt with an apparently different content area. However, each problem was formatted identically, and consisted of the following elements. The first element was a brief cover story that identified five hypothetical students by first name, and described two kinds of everyday characteristics or variables about them as a group. As in Experiment 2, one variable was categorical (e.g., it might be the residence hall in which a student might live, or his or her major); the final variable was the scalar variable (e.g., it might be the number of hours each student had earned toward graduation, or the starting salary of the student's first job). The second element, or text, was a set of nine factual premises, each of which established either an "is" or an "is not a" relationship between a value on one of the variables and one or two values on a second variable. For example, a premise might assert that "Frank" (a value on the Person variable) "earns $47,000" (a value on the Scalar variable). A space for the participant's answers was the next element of each logic table problem. In the answer space were three columns labeled with that problem's variables (i.e., the Person, Categorical, and Scalar variables), and five blank lines below them. Each problem appeared by itself on an 8.5 x 11 in. (21 x 27.5 cm) sheet of paper.

Varying the structure of the logic table problems. For each of the three cover stories, three different versions of the logic table problems were created. These different versions were either Isomorphs or Different Structures, compared to the other versions.

Logic table Isomorphs were created by having text that established one-to-one consistent mappings across the set of the nine factual premises and across the set of three contents. For example, in the first problem, which had to do with students, their residence halls and their hours toward graduation, the initial premise given was "Alicia does not live in Jasper." In the second problem, which had to do with a different set of students, their job titles after graduation, and their salaries, in the isomorphic condition, the initial premise was "Frank is not the Sales Manager." In this Isomorphic condition, the value of the Person variable "Frank" is completely equivalent with the value "Alicia," and this was true for each of the values mentioned in the problem's text. In the Different Structures condition, a participant who may have seen the "Alicia-Jasper" statement in the first problem might see an initial statement like "Hayden is not the Banker" in the second problem. The character "Hayden" does not play the same role, in terms of his connected values, that Alicia plays in the first problem. Moreover, none of the other values mentioned in the second problem correspond to any of the values used in the first problem. All versions of all problems had identical correct answers.

Each logic table problem was attached to an answer grid. This grid showed each of the five values for one variable (such as "student's name") on axis, and each of the five values for another variable (such as "student's major") on the opposite axis to create a box of 25 cells. The grid contained three such boxes, each box linking two of the problem's three variables, for a total of 75 cells. These 75 cells represent the total number of ways in which the values of the three variables (Person-Categorical-Scalar) could be connected (e.g., "Brandon-Economics major"; "Economics major--39 hours toward graduation"; and "Brandon--39 hours toward graduation").

The participants were invited to use the grid by putting an "X" into any cell representing a connection that was explicitly ruled out by one of the problem's statements, as in the example "Alyssa is not a Psychology major," or in any cell that was ruled out by an inferential process, as in the example, "Emma has earned fewer than 39 hours toward graduation," which rules out cells in which Emma was connected with greater than 39 hours. Further, the participants were invited to put a check mark into any cell whose connection was either directly stated in the problem, as in the example, "Daniel lives in Felton Hall," or if the connection was derived from other information. For example, if, through a process of deduction, any row or column of a box came to contain four Xs, then the remaining only open cell in that row or column must be assigned a check mark. Further, once a check mark had been assigned to a connection, then any remaining open cells in the rows or columns that intersected with that connection could be Xed out. For example, if it is established that Daniel lives in Felton [Hall], then he cannot also live in any of the other four residence halls, nor can anyone else live in Felton.

By looking at the pattern of Xs and check marks in any particular box of 25 cells (such as the Persons-Residence Hall box), and by coordinating this pattern of Xs and check marks with the pattern in another box of 25 cells (such as the Residence Hall-Hours Toward Graduation box), it is possible to construct a "reasoning path" that solvers might take as they worked their way through the problem. Depending on the information that is given in the problem's initial statements, coordinating the check mark and X pattern in each box of 25 cells creates a sequence of inferences that become available earlier or later in a chain of inferences as reasoners work their way through the problem. This concept of a deductive pathway became the basis for the structural manipulation used in this experiment.

A reasoning algorithm was developed to establish the deductive pathway for each problem. This algorithm operated in three stages. In the first stage, each of the problem's textual statements was coded onto the answer grid to create starting point for the conditional reasoning processes that could be brought to bear. In the algorithm's second stage, each of the three boxes was examined to see if any row or column contained four Xs. In that case, a check mark was put in the remaining place in that row or column, and all remaining open cells in the intersecting row or column were then Xed out. In the third stage, rows and columns in a particular box were examined to see if any contained three Xs. If there was more than one row or column that had three Xs, one of them was chosen randomly. For the row or column that was chosen, the rows and columns of a different box were examined to see if conditional reasoning processes could produce an inference. For example, suppose that, after coding all the problem's initial statements into the answer grid, the box showing the linkages between the Persons and the Categorical variable, and the Persons and the Scalar variable, showed that a particular name was definitively not linked to three values on the Categorical variable, and not linked to three values on the Scalar variable as follows:

[P.sub.1] [not equal to] [C.sub.1], [C.sub.2], [C.sub.3] [P.sub.1] [not equal to] [S.sub.1], [S.sub.2], [S.sub.3]

If the box showing the linkages between the Categorical variable and the Scalar variable shows the following connection as definitely checked:

[C.sub.1] = [S.sub.4]

Then a conditional reasoning process might occur:

If [P.sub.1] [not equal to] [C.sub.1] and, [C.sub.1] = [S.sub.4] then, [P.sub.1] [not equal to] [S.sub.4]

And an X can then be placed in the [P.sub.1]-[S.sub.4] linkage. Once this X has been placed, then the algorithm can go back to its second stage because P1 has now been Xed out for four values, [S.sub.1], [S.sub.2], [S.sub.3], and [S.sub.4]. There is only one value of the scalar variable remaining, and so [P.sub.1] must be associated with it:

[P.sub.1] = [S.sub.5]

And a check mark can now be placed in the box for that linkage.

All of the versions of the three logic table problem had the same answers, but for the problems that were structurally different from each other, the answers were logically attainable in different sequences. This algorithm provides a way calibrating that sequence by indicating, for each correct answer, the point in the sequence at which there was now enough information to enable that answer to be deduced.

Finally, in addition to the structural manipulations described above, some of the rows and columns of the problems of the boxes for the Person variable and Categorical variable were transposed for each problem. This was done so that when the answer grid was correctly filled out, it wouldn't look the same for consecutive Isomorphic problems, thus eliminating a potential purely visual memory strategy for solving the problems.

Design The reasoning pathway manipulation described above was the basis for creating three problems whose reasoning pathways were substantially different from each other, in the sense that the order in which deductions could be made was different. The difference in reasoning pathways was the structural manipulation in Experiment 3. Each of these three structures was crossed with each of the three cover stories that were used. Participants were randomly assigned to either an "Isomorphs" condition in which they were asked to solve three logic table problems whose reasoning pathway was identical, or to a "Different Structures" condition, in which they were asked to solve three logic table problems that had different solution paths.

Procedure In addition to the instructions as provided in the procedural overview section, the participants were instructed to deduce the identity of each person, and their values on the categorical and scalar variables. The "answer sheet" displayed blank spaces for each of the five values for each variable.

Results

Fifty-eight people participated in the study. Of these, 10 concluded their participation before they had completed their deductions on the third critical reasoning problem. One participant concluded his participation before he had completed his deductions on the second critical reasoning problem. For the 10 participants who did not complete the third problem, their mean time to completion through two problems (M = 35.7 min) was not different from the mean time to completion for two problems for those who completed the study (M = 35.2 min). These eleven participants' data are excluded from the following analyses. There were 21 remaining participants in the Different Structures condition and 26 participants remaining in the Isomorphs condition.

Solution Time Effects There was a main effect of solution attempt number on solution time, F(2, 90) = 6.08, p < .001, [[eta].sup.2.sub.p] = 119. That is, for reasoners who attempted and completed all three problems to their satisfaction, the reduction in solution time from Problem 1 (M = 1021 sec, SD = 395) to Problem 3 (M = 691 sec, SD = 484) was significant, and large (32%). This main effect is not necessarily incongruent with the DGC position, which predicts that the reasoning problem's representation is strengthened by repetition, possibly leading to a speedup in solution time across a series of reasoning problems. However, the DGC prediction is based on a repetition of structure across the series of problems. Consequently reasoners who solved Isomorphs should be expected to show faster solution times than did reasoners who were in the Different Structures condition. However, this effect was not observed, F(1, 45) < 1, p > .05. Finally, it could be the case that, even if reasoners in both the Different Structures and Isomorphs conditions showed a speed-up in solution time, it might be that the effect was more pronounced for reasoners solving their second and third Isomorphic problem, compared to reasoners solving their second and third Different Structures problem. This would suggest that an interaction of solution attempts X subject type (Isomorphs vs. Different Structures) may lend support for the DGC position. However, this possibly expected interaction of subject type (Isomorphs vs. Different Structures) X solution attempts (Problems 1, 2, and 3) was not significant either, F(2, 90) < 1, p > .05. Reasoners solving logic table problems that had Different Structures sped up in subsequent solution attempts at the same rate as did reasoners solving Isomorphs. These effects are shown in Figure 3.

Accuracy Effects The logic table problems were scored by awarding one point for each of the three types of connections (Person to Categorical, Categorical to Scalar, etc.) for each of the five hypothetical student names in the problem, thus yielding a maximum score of 15 points for each problem.

As Figure 4 shows, the reasoners' deductive accuracy improved rather dramatically across the set of three logic table problems, F(2, 90) = 10.91, p < .001, [[eta].sub.2.sub.p] = .195. The participants averaged 7.55 correct deductions on their first attempt (SD = 4.46), and this went up to 10.51 (SD = 4.73) by their final attempt, an improvement of 39%. As with the solution time main effect, this effect is not necessarily incongruent with the DGC position, which predicts that the reasoning problem's representation is strengthened by repetition, possibly facilitating improvements in accuracy across a series of reasoning problems. As with the solution times, the DGC position predicts that improvements in accuracy are a function of structural repetition, and so that effect should be limited to reasoners who are solving a series of Isomorphs. However, this effect was not observed, F(1, 45) = 1.54 p > .05. Reasoners solving a series of problems that had different underlying structural relationships (i.e., different pathways) did not have less deductive accuracy than did reasoners solving a series of isomorphs.

Finally, it could be the case that, even if reasoners in both the Different Structures and Isomorphs conditions showed similar increases in deductive accuracy, DGC theory predicts that the improvement in accuracy would be greater for reasoners solving their second and third Isomorphic problem, compared to reasoners solving their second and third Different Structures problem. As with the solution time effect, this reasoning would suggest that an interaction of solution attempts X subject type (Isomorphs vs. Different Structures) may lend support for the DGC position. However, this expected interaction of subject type (Isomorphs vs. Different Structures) X solution attempts (Problems 1, 2, and 3) on deductive accuracy was not significant either, F(2, 90) = 1.04, p > .05. Reasoners solving logic table problems that had Different Structures showed the same rate of improvement in deductive accuracy as did reasoners solving a series of Isomorphs.

Depth of Inferences One of the claims in this experiment is that the structural manipulation in this study resulted in reasoning problems that either shared, or did not share, a reasoning pathway that resulted in corresponding inferences being made at similar or different points in a sequence, depending upon whether the reasoner was faced with Isomorphs, or problems with Different Structures. Evaluating this claim involves showing that the Isomorphs did indeed share more aspects of their reasoning pathway than did the problems with Different Structures. To test this claim, the reasoning algorithm referred to above was run for each of the logic table's three structural variations on each of the three content areas (a total of nine different logic table problems). For each problem, the order in which each inference was available to be made was noted. There were 15 correct answers for each problem; two correct answers were always given in the problem's text. These 15 answers were assigned a specific "depth" which was determined by its place in the sequence that the reasoning algorithm indicated that the inference was available to be made. For example, the first inference was given number 1, the second number 2 and so on. In this way, each correct answer was assigned a depth from 0 (for the two answers in each problem that were simply stated to 13 (for the final inference that became available to be made according to the algorithm. After the algorithm was run for each logic table problem, the depth numbers were arranged in a one-dimensional string to create a "depth vector" for the answers for that particular problem. If the Isomorphs actually do share more of a common reasoning pathway, then these depth vectors should be more highly positively correlated for Isomorphic logic table problems then they are for the logic table problems with Different Structures. As Table 2 shows, this outcome is basically supported, the depth vectors of the Isomorphs are more positively correlated than are the depth vectors of the logic table problems with Different Structures. Regardless of the content of the cover story, and the names given to the particular variables in a logic table problem, the results of the analysis, which are based on an algorithm that could be very plausibly used by the reasoners themselves, indicated that the inferences of the Isomorphs would tend to become available in approximately the same sequence, to a greater extent than was so for the problems with the Different Structures.

This leads to a prediction about the participant's ability to correctly make these inferences. The results of the algorithmic analysis suggest that, if the participants are building up a representation of the reasoning problem, then they should be more likely to correctly make "deeper" inferences (inferences occurring late in the sequence) in repeated Isomorphic problems than they would be for deeper inferences in the problems with Different Structures. This prediction was tested in a two (Different Structures vs. Isomorphs) X three (repeated attempts on logic table problems) ANOVA using both deep inferences (mean score on inferences numbering 11, 12, or 13) and shallow inferences (mean score on inferences numbering 1, 2, or 3) as the dependent variables.

As shown in Table 3, with regard to the effect of repetition on the shallow deductions, there was a small, but statistically significant, effect, F(2, 90) = 3.26, p < .05 [[eta].sup.2.sub.p] = 068. On their first logic table problem, the participants averaged 1.66 (SD = 1.15) shallow deductions correct (out of three possible). By their third logic table problems, the participants were doing better, averaging 2.15 (SD = 1.12) correct shallow deductions, approximately 23% better. Congruent with the DGC position, the effect of repetition was not moderated by inclusion in the Isomorphs or Different Structures condition, F(1, 45) = 3.32, p = .075, and the repetition X structure (Different Structures vs Isomorphs) was not significant, F(2, 90) < 1.

There was also a substantial effect of repetition on deductive accuracy for the deep inferences, F(2, 90) = 7.76, p < .001, [[eta].sup.2.sub.p] = .147). Reasoners averaged 1.32 (SD = 1.23) deep deductions (out of three possible) on their first attempt, and this improved to 1.94 (SD = 1.15) by the third logic table problem, about a 47% improvement. However, contrary to the DGC position, this effect was not moderated by inclusion in the Isomorphs or Different Structures condition, F(1, 45) = .283, p > .05. Perhaps more importantly, the repetition X structure (Different vs Isomorphs) interaction was also not significant, F(2, 90) = 1.70, p > .05, with the trend going against the prediction of the DGC position. The reasoners solving logic table problems that had Different Structures actually did marginally better than did the reasoners in the Isomorphs condition at making the deep inferences correctly on successive problems.

Discussion

Experiment 3 represented an attempt to organize the structure of the reasoning task in terms of the chain of inferences making up the reasoning pathway. However, despite this different organizational structure used Experiment 3, the overall pattern of results regarding the solution times for successive logic table problems seemed remarkably similar to that seen in Experiment 2. In Experiment 3, as in Experiment 2, solution times declined substantially across a series of logic table problems, and the amount of time saved was the same for reasoners in the Different Structures and in the Isomorphs conditions. Moreover the rate of change appears to be the same for reasoners in both conditions. In Experiment 3, deductive accuracy finally fell from the ceiling levels seen in Experiments 1 and 2. When the elusive accuracy effects finally manifested themselves, they were similar to the solution time effects. The reasoners' deductive accuracy improved substantially across the series of three logic table problems. However, as with the solution time effects, the accuracy results did not provide any evidence for the DGC position. The improvement in deductive accuracy was not moderated by the group that the reasoners were in (Different Structures vs. Isomorphs). Looking at the depth of the deductions shows that the improvement in deductive accuracy was not restricted to either the "front" (shallow, and readily available deductions) or the "back" of the problems (deeper, farther along the reasoning pathway, and requiring a longer chain of inferences). The reasoners in both Deep Structures and Isomorphs conditions improved generally on both portions of the reasoning pathway. Although there are some specific findings that are congruent with the DGC position, the overall pattern of the results is not particularly supportive, and still less so for the dual-systems position, creating a challenge to interpret the findings.

If the pattern of results (with regard to both speed and accuracy) is not dependent on the structural variation the reasoners found themselves in, then the results could be dependent on some other external factor, such as the emergence of a reasoning strategy. Although this term seems to have a number of meanings with regard to reasoning, Evans (2000) has argued that neither the mental models theory (Johnson-Laird & Byrne, 1991), nor the mental logic theory (Rips, 1994) have any stipulations that would prohibit the concept of a reasoning strategy. It is also widely known that in reasoning tasks that are extended over time (such as these logic table problems) people at least seem to search for strategies (Roberts, 2000). Finally, Johnson-Laird, Savary, and Bucciarelli (2000) showed that there can be a visual, written down component to the strategies that a reasoner might take in evaluating lengthy strings of verbal information, such as might occur for a solver in these logic table problems.

GENERAL DISCUSSION

Given the number of studies that have explored human reasoning, there is surprisingly little known about what happens when people make several engagements with the same type of reasoning problems within an hour-long time frame, so these studies are interesting for the light they may begin to shed on those processes. Specifically, Experiment 1 (Repeated reasoning over problem isomorphs) showed that people sped up substantially in their solution times of successive isomorphic logic table problems, a result predicted by the DGC position. Experiment 2 (Changing the pattern of connections among the variables) replicated that finding. However, the results of Experiment 2 also showed that reasoners showed a comparable speed-up in performance across a series of logic table problems that did not share a structural representation for reasoning, a result not congruent with the DGC position. Experiment 2 further showed that the presence of the logic table answer grid did not have any effect on accuracy or solution time. Dual-systems theories predict that the grid should have made the reasoning process more deliberative, more "analytic" and therefore probably slower than reasoning in the absence of the grid, based on a number of studies that have manipulated instructional effects (e.g., Evans, 2000). But actually when the grid was present, the solution times were marginally, but not significantly, faster than when the grid was absent. This is probably because almost 100% of the participants in the logic Grid Absent condition made up at least a rudimentary grid of their own. Experiment 3 (Different pathways through the matrix) replicated the effects of Experiment 2 with regard to solution time effects for Isomorphs and for problems with Different Structures. Experiment 3 also showed that deductive accuracy improved across the series of three problems, and this effect was not moderated for reasoners in the Different Structures conditions. Similarly, the improvement in deductive accuracy was comparable for both "shallow" inferences (those that were more readily available earlier in the chain of inferences required to complete the problem as well as "deep" inferences (those occurring at the end of the chain of inferences, according to the plausible algorithm that was developed to generate an inferential path). This improvement in both shallow and deep inferences was not moderated by inclusion in the Different Structures condition. That is, even participants who solved logic table problems that had weakly correlated reasoning pathways were just as likely to get to the "back" of such problems as were the reasoners who solved logic table problems that shared reasoning pathways, a result that is not predicted by either the DGC or the dual-systems position.

Some, but not all, of these findings are accountable within the DGC framework (Osman, 2004), which itself should be understood as an alternative to a commonly-held dual-systems view (Evans, 2008, 2009). Osman has argued that at least two of the phenomena seemingly explained by the dual-systems view, namely that of speed and ease on some types of reasoning tasks, and the concomitant absence of a verbalizable reasoning protocol during such tasks, could be explained without resorting to a dual systems view. Instead, Osman suggests that changes in the underlying representation supporting the reasoning can be understood as a form of learning, and therefore can be captured in a production system architecture such as ACT-R (Anderson & Lebiere, 1998; Taatgen & Anderson, 2002) designed to model performance on learning tasks. This approach seems plausible, especially given the link between the formal properties of the production system (i.e., the condition-action pairs) with the typical sentential format in which conditional or hypothetical reasoning is expressed ("if/then" statements).

Two Systems or One?

The distinction between the processing done by the fast heuristic system and by the slow, effortful, analytic system for reasoning has been supported in quite a few studies (see Evans, 2008, for a review). In addition, correctly wielding the analytic system is apparently related to a number of other cognitive factors, including a possibly intrinsic ability level. However, Osman suggests that what might differentiate any two groups of reasoners--a fast heuristic group vs. a slow, analytic group is not necessarily any difference in systems or even intrinsic differences in ability, but rather a difference in the number of production rules in their systems, and their complexity (which might be a processing difference rather than an ability difference). How many production rules one builds might be a function of what one wants to do with the task rather than an inherent limitation on ability. So, for example, asking people to come up with the number of counterexamples to the initial conclusion generated and evaluated in reasoning situations predicts "correct" performance over and above simples measures of cognitive ability. Differences in reasoning correlate with cognitive ability measures, but this can be interpreted as showing differences in the characteristics of the underlying representation supporting the reasoning, rather than the kind of reasoning used (Osman 2004). This explanation casts the findings that have been used to support the distinction between the heuristic and analytic systems into a different light. For example, Stanovich and West (2000) showed that in in tasks where the two systems cue different responses, high SAT scores (a measure of cognitive ability) were correlated with the use of slow analytic processes. By contrast, lower SAT scores were correlated with performance errors that suggested those individuals were reliant on fast heuristic processes. Findings such as these have been supported in a number of other studies (e.g., Klaczynski, Gordon, & Fauth 1996). The slow processes were also very cognitively demanding, requiring executive functions and workload capacity, resulting in a course of processing that was explicit, or could be made explicit. The fast heuristic processes did not seem to make these demands, and did not seem to leave an explicit trace of their operation. The DGC framework proposes that the different types of reasoning (implicit, explicit and automatic) are dependent on the representations from which the participants reason, and this is not reliably differentiated according to the demands made on the central executive. For example, the DGC proposes skilled reasoners demonstrate a type of automatic reasoning that is dependent on well-formed representations of highly specialized rules; these have acquired strength and become attuned to the circumstances in which they apply. Comparisons between poor and skilled reasoners suggest that in any given task, some demonstrate knowledge that is highly familiarized and results in rapid responses (Galotti et al. 1986). Such reasoners have gone beyond both the implicit and the explicit stages.

Although the DGC perspective can thus account for the speed-up in response time seen for participants solving Isomorphs in the three experiments without requiring the work of either a heuristic or an analytic reasoning system, it cannot seemingly account for the findings that saw participants in the Different Structures conditions in Experiments 2 and 3 show an almost identical speed-up in response time compared to those participants in the Isomorphs conditions. However, it may be the case that the experimental manipulations on the logic table problems used in Experiments 2 and 3 actually did not create the need for structurally different representations. That is, it is not clear whether the structural manipulation was really that effective in directing the subject into a different reasoning pattern or format. For example, to consider the manipulation used in the materials in Experiment 2, participants in the Isomorphs condition solved problems that had the same pattern of connectivity among the variables. For a problem whose structure was [PCS], this meant that the participant would always see statements in the problem's text that connected a value on the Person variable, (i.e., a name) with a value on the Scalar variable, and statements that connected a value on the Categorical variable with a value on the Scalar variable, or [P-S, C-S]. This would not be the case for participants in the Different Structures conditions. These participants might see a sequence of logic table problems having the following structures [PCS], [CSP], [SPC], and so the statements they would see would have the following pattern of connectivity: [P-S, C-S]; [C-P, S-P]; and [S-C, P-C]. In other words, participants in the Different Structures condition in Experiment 2 never saw the same "kinds" of statements repeated in their problems' texts. However, even the modest similarity that remains among the logic table problems with Different Structures may be sufficient to produce a similarity in representational structure that is sufficient to produce the speed-up seen in the Different Structures conditions in Experiments 2 and 3. If so, then this objection raises questions about what would be required in the set of logic table problems to create representations that did not share any structural similarity.

The experiments reported here are based on assumptions that the reasoners in the three studies were all starting from the same relatively low baseline regarding a representation for reasoning in logic table problems, and that the reasoners would build whatever representation they used on the fly in the experiment. Based as it is on a model of implicit learning, the DGC implies that the time course involved in building a representation for reasoning might be longer than the time that the participants typically spent with the materials in these three experiments. Hence, for college-educated participants solving what may be for them relatively simple problems like these, the structure for reasoning may have already been built before coming in the experimental situation. That might explain why the participants sped up as much on the problems with Different Structures as they did on the Isomorphs, but it raises the problem of why such participants would show any speed-up effect at all on the problems that were Isomorphs. If the representation for reasoning had already been effectively built prior to their engagement with the materials in the experiment, then no speed-up in performance should be expected for participants solving problems in the Isomorphs conditions because their representations for reasoning were not really increasing in strength, stability, or distinctiveness as the DGC claims during the time frame of the experiments.

There are some limitations present across the three experiments that limit their interpretation. First, there was a great deal of racial and age-homogeneity across the set of participants in these studies. For example, almost all of the participants were between 20 and 24 years of age. Second, the time-intensive nature of the studies limited the sample size. Both of these factors have an effect on external validity. Using response time as a measure has implications as well. For example, it is well-known that outliers can have an influence on the results. There was no attempt made in the present studies to screen for outliers, because here, the existence of outliers may actually be part of the effect. If the dual-systems perspective is true, the people who look like outliers might be the ones who are really trying to reason analytically, and doing so, might really slow them down. An analysis of the kurtosis and skew of the response time distributions can shed some light on these issues. Consistent with the dual-systems interpretation, kurtosis and skew measures were generally between 0 and 1.5 for each response distribution in Experiments 1 and 2. These numbers indicate that the distributions were slightly more platykurtic than normal, and slightly more positively skewed. It was only on Problems 2 and 3 of Experiment 3 in which the skewness measure went up to +3.5 or +4, indicating substantial positive skew (i.e, a few participants spending a long time on the problems) This is not surprising, given that the mean response times in Experiment 3 were substantially longer than they were for any of the problems in the first two experiments, indicating that these problems were substantially more challenging than those in Experiments 1 and 2.

The field of cognitive science is grappling with the nature of the representations required for reasoning, the conditions that may serve as the input for one or another system, or even more fundamentally, whether two such systems are required. For people outside the field of reasoning research, such as curriculum designers and planners working in higher education who are attempting to foster critical thinking skills, among which, successful deduction must surely be required, it may be necessary to wait until some answers to these and other questions have been clearly established. Until the components of reasoning, and which, if any, seem to require are more clearly established, and attempts to develop curricula to enhance critical thinking may be in vain. Until then, the Dynamic Graded Continuum position remains an intriguing alternative to the dual-systems viewpoint, even as it raises persistent questions about the nature of the representation required for reasoning, its contents, and the time course involved in building it.

APPENDIX A

AN EXAMPLE OF THE LOGIC TABLE PROBLEMS USED IN EXPERIMENT 2

The students graduate and get jobs. Alyssa, Bob, Carrie, and Denny are students who graduate and each student gets one of the following jobs: chemist, librarian, banker, and writer. Their salaries are $45,800, $50,200, $42,500, and $37,400. The following facts are all true:

Alyssa is not a chemist or a banker.
The person who earns $37,400 is not chemist or a banker.
Bob is not a banker or a writer.
The person who earns $45,800 is not a banker or a librarian.
Alyssa is not a writer.
The person who earns $42,500 is not banker or a writer.
Carrie is not a banker or a librarian.
The person who earns $37,400 is not a writer.

What is each person's job and salary?

Person     Job     Salary

Alyssa     --        --

Bob        --        --

Carrie     --        --

Denny      __        __

[P S C]


Author Note: I would like to recognize and thank the following people who worked on the materials, or as the experimenters for these studies (in chronological order of their involvement): Jacqueline Wade and Megan Crites (Experiment 1); Amy Charlton and Koyeli Sengupta (Experiment 2); Jacob Nidey, Christine Lowell, Keith Huddleston, Sonia Shah, and Amy Leonard (Experiment 3).

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John Best

Eastern Illinois University

Author info: Correspondence should be sent to: John Best, Department of Psychology, Eastern Illinois University, Charleston, IL 61920. Email: jbbest@eiu.edu

TABLE 1 Structure of Logic Table Problems Used in Experiment 2

Display Variable   Nondisplay Variable   Nonconnected Variable
Person             Categorical           Scalar
Person             Scalar                Categorical
Categorical        Person                Scalar
Categorical        Scalar                Person
Scalar             Person                Categorical
Scalar             Categorical           Person

TABLE 2 Correlation of Depth Vectors for Three Different Problems &
Three Different Structures

              Structure 1
              Problem

              1             2     3
          1   -
Problem   2   .62           -
          3   .28           .66   -
              Structure 2
              Problem
              1             2     3
          1   -
Problem   2   .48           -
          3   .68           .63   -
              Structure 3
              Problem
              1             2     3
          1   -
Problem   2   .84           -
          3   .87           .92   -

              Problem 1
              Structure

              1             2     3
          1   -
Structure 2   .31           -
          3   -.12          -.07  -
              Problem 2
              Structure
              1             2     3
          1   -
Structure 2   -.23          -
          3   -.08          -.42  -
              Problem 3
              Struct        ure
              1             2     3
          1   -
Structure 2   -.27          -
          3   .17           -.24  -

TABLE 3 Shallow and Deep Inferences on Logic Table Problems
in Experiment 3

Inference Depth   Problem 1      Problem 2       Problem 3

Shallow           1.65 (1.15)    1.83 (1.22)     2.15 (1.12)
Deep              1.32 (1.23)    1.94 (1.15)     1.94 (1.15)


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