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I ’,. .. .n- a. - v. -u . a —-- ‘,,......_. .y'»o.'. . a. ‘ , _ _ . . . . ...— a- ..... . . 4.,... u. 09" -b -' . -‘P-o‘/u'- s-r a.-.‘ no—o- I . o. 0.9- - ‘ o -u .- . . ,. o .n .' .-r' 0-. . “to, ... .._ ""['(I]’.""‘1'."‘-““"""‘!."- I 4o.’\--" " ' -.~ .a - . - ‘ ... ._.. .. 4.7‘0‘.“.....‘-n—c g ' o .g. LIJRARY Mic! igan State Univcrsity ”a! magnum BY “ T Hons & SflNS’ WERNER"?- ABSTRACT EFFECTS OF TWO TYPES OF TRAINING AND PROBLEM STATUS 0N SYLLOGISTIC PERFORMANCE By David William Carroll The major intent of the present study was to produce differ- ences in syllogistic performance as a function of training techniques and problem types, and then to assess both the depth and the scope of these differences. Forty-two introductory psychology students were exposed to either a spatial or an algorithmic treatment condition during a l6- problem training session. They were then tested on 32 problems, dif- fering in specifiable characteristics. The results indicated that both experimental groups performed at a higher level than the control group which received no training, yet the effect was restricted to those problems that do not have a valid logical conclusion (indeter- minate problems). Though there was no treatment effect on determinate problems, post-hoc analysis revealed a significant effect on two types of indeterminates. The results were discussed in terms of behavioral rule con- siderations. An analysis of the results was also performed in terms David William Carroll of the underlying skills of verification and falsification of logical propositions, and these skills were specified as several plausible rules. EFFECTS OF TWO TYPES OF TRAINING AND PROBLEM STATUS 0N SYLLOGISTIC PERFORMANCE By David William Carroll A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF ARTS Department of Psychology ACKNOWLEDGMENTS I am pleased to express my appreciation to Dr. Donald M. Johnson, chairman of my committee, for his guidance and assistance in the preparation of this thesis. Gratitude is also extended to both Dr. Andrew Porter and Dr. Gordon Wood for their contributions to the design and execution of the work. ii TABLE OF CONTENTS Page INTRODUCTION ........................... l A. The Syllogism ....................... 3 B. Historical Introduction .................. 3 METHOD .............................. l0 A. Subjects .......................... 10 B. Procedure ......................... l0 C. Hypotheses ......................... l6 RESULTS ..... I .~.— ..................... l8 A. Training results ..... . ................. 18 B. Major findings ....................... 19 C. Treatment and Treatment-Measures Effects .......... 23 D. Measures Effect ...................... 34 DISCUSSION ............................ 43 A Treatment Effect ......... Tb ............ 43 B. Treatment-Measures Interaction ............... 44 C. Measures Effect ...................... 48 D Conclusions ........................ 51 E Implications for Further Research ............. 52 LIST OF REFERENCES ........................ 53 APPENDICES A. Training Booklets ..................... 55 B. Training Items ....................... 6l C. Test Items ......................... 65 D. Rule-Exception Results per Problem ............. 72 iii Table A CDNO‘LH 10. ll. 12. l3. 14. IS. l6. I7. 18. LIST OF TABLES Scores in percentages on each of l6 training problems . . . Figure, order and premise combination comparisons total number correct per group (n=l4) per problem . . . . Total percentage scores of each group on each problem Total percentage scores of each group on each problem dimension ....................... Analysis of variance using original data ........ Analysis of variance using transformed data ....... Analysis of variance using total score data ....... Fourteen two-way ANOVAs and Scheffé contrast confidence intervals ....................... Analysis of variance using determinate problems ..... Analysis of variance using indeterminate problems . . . . Total percentage scores of each group on each category Analysis of variance for "see rule" ........... Analysis of variance for "correct rule" ......... Analysis of variance for "correct exception" ...... Analysis of variance for “see exception" ........ Between-problem ANOVA .................. U-M ANOVA for determinate problems ........... A-M ANOVA for determinate problems ........... iv Page 18 20 21 22 24 24 24 25 28 28 32 33 33 33 38 Table Page 19. U—M ANOVA for Group B problems ............... 4O 20. A-M ANOVA for Group 8 problems ......... ' ...... 4O 21. U-M-P ANOVA for Group A problems ..... i ......... 41 22. A-M-N ANOVA for Group A problems .............. 41 LIST OF FIGURES Figure Page 1. Properties of the syllogism ................ 2 2. Dominant response chart .................. 15 3. Profiles of 14 scores for each group ........... 27 4. Per cent correct response, per group, on three dimensions of problems ................. 29 5. Rules and exceptions ................... 31 6. Per cent correct response, per group, on three types of problems .................... 35 7. Per cent correct response, per group, on determinate rules and group B indeterminates ............ 36 8. Per cent correct response, on indeterminate problems . . . 37 vi INTRODUCTION This thesis was an attempt to train subjects to improve on their syllogistic performance, with the goal in mind of assessing the depth and scope of any improvement such training might provide. There are undoubtedly many ways of effecting an improvement in any perfor- mance, but my aim has been to closely examine but two methods of train- ing, in terms of the quantitative and qualitative changes they have produced here, and in terms of both the positive and negative aspects of these changes. Inasmuch as some details of the syllogism will be employed in the discussions throughout this thesis, a brief presenta- tion of the terminology to be used will be of merit. A. The Syllogism The syllogisms used here consist of two premises, with five possible conclusions. Each statement of the syllogism is selected from four possible logical propositions (see Figure l on next page). When a set of premises is used in a description, the first premise is mentioned first, such as in AE or "All X are Y, No Y are Z." The set of two premises may also be described by "averaging" the dimensions of quantity and quality, thus forming a single description. To take the same example, the A premise is universal affirmative, and the E premise is universal negative, so that the AE is universal and mixed. As used here, the quantity dimension indicates the extent of reference, and has 1 the values of universal, mixed and particular. The quality dimension indicates the form of reference, and has the values of affirmative, mixed and negative. It is the latter description of the problem as a universal mixed, or UM, and not as an AE, that will be employed more extensively. This is so because, for one, it is the latter variables that will be manipulated in this experiment and, secondly, there is an economy of reference when employing this system. For instance, there are 4 AE problems, but only two corresponding UM problems in all. There are 64 problems (16 premise combination, each in 4 figures) but only 16 problem types. Propositions: Terms: A, B, X, Y, etc. A = All A are B (universal affirmative) E = No A are B (universal negative) Premise combinations: I = Some A are B (particular affirmative) A+E=AE or universal mixed 0 Some A are not B (particular negative) I+E=IE or mixed mixed Figures: lst--All X are Y 2nd--All A are B All 2 are X All C are B All 2 are Y (valid) All C are A (invalid) 3rd--All J are K 4th--All P are Q All J are L All Q are R All L are K (invalid) All P are R (valid) Some L are K (valid) Some R are P (valid) Problem status: lst letter--signifies determinate or indeterminate 2nd letter--signifies universal, particular or mixed 3rd letter--signifies affirmative, negative or mixed Thus, the premise combination AA, in the first figure (AA-1), is a determinate universal affirmative (DUA). AA-2, however, is indeter- minite (IUA). Figure 1. Properties of the syllogism. The terms of the syllogism, as indicated above, are simply the elements that are being referred to. Their importance in the present study is negligible, for only letters of the alphabet are used as terms. The four figures of any given syllogism leave unaltered the basic information (quantity and quality of reference), but may change the determinateness/indeterminateness (whether or not a problem has a solution) of the syllogism. When referring to a particular figure of a premise combination, the notation 10-3, EA-4 and so on will be used. Finally, it should be noted that the status of a problem is but an ordered set of three properties of that problem; the first property, however, differs from the other two in several ways. The quantity and quality dimensions are straightforward "aver- age" of the two premises, but the determinateness of the problem is a property of the whole syllogism, not a combination of its parts. Fur- thermore, it is contingent upon many factors--including the other two dimensions, as well as the figure of the problem, the order of the premises, and the premise combination. This point will be considered again later, for though a logical principle (e.g., undistributed middle term) is available to account for the determinateness of a problem, the psychological dimensions underlying an individual's recognition of a determinate item are still in need of clarification. B. Historical Introduction Psychological investigation of logical problems and the reason- ing process began in the 19305. The type of research carried out seems divisible into two rather large classes. One type of work has been theory-laden, with the principal intent of the investigation being to obtain supportive evidence for one or another of the theories of the reasoning process that have been offerred in the last forty years. The other, more theoretically innocent line of research has been specific in nature, attempting to delineate details concerning the difficulty of certain problems, the tactics experimental subjects resort to in trying to solve the problems, and the properties a training procedure must have in order to be successful. I will want to acknowledge the contri— butions to this study, of both lines of research before outlining the purpose behind my own work. Woodworth and Sells (1935) outlined the first psychological hypothesis of the reasoning process, by noting that their subjects (Ss) seemed to process the material non-logically. They termed the "atmos- phere" error the tendency for an §_to choose an A response to AA syllo- gisms, an E response to EE syllogisms, and so on, without respect to the logical relations involved. A different post-hoc analysis of errors, by Chapman and Chapman (1959), supports the view that a primitive type of reasoning is respon- sible for these atmosphere errors. Their hypothesis of "probabilistic inference" is hinged on the finding that §s apparently try to convert propositions such as "All A are B" into "All B are A" (an invalid con- version), then deal with the problem appropriately after that point. Henle (1962) and Ceraso and Provitera (1971) have argued similarly. Henle (1962) argues that an error on a logical item does not necessarily implicate a non-reasoning problem-solving process, and that errors such as slipping in probabilistic yet non-implicated additional premises in an argument may account for these supposed failures to reason. Ceraso and Provitera (1971) modified traditional syllogisms in order to test the hypothesis that premise misunderstanding is the basic component of a high error rate. Their results suggest that errors previously attri- buted to the atmosphere effect might well be due to premise misunder- standing, with the subject reasoning properly from that point on. To experimentally assess the two hypotheses, Simpson and Johnson (1966) designed two training conditions, one anti-atmosphere and one anti-(invalid) conversion. The superficiality of the atmosphere effect is exposed by the fact that it diminishes greatly even after brief training. Anti-conversion training was somewhat less successful, but the two interpretations appear to overlap by accounting for the same error in two different ways. In a more recent study, Johnson (unpub- lished) found an interesting effect, that both of these types of train- ing reduce only the indeterminate errors and that, in fact, the control group did slightly better than the experimental groups on determinate syllogisms. Ceraso and Provitera's (1971) training effect likewise produced larger differences on the indeterminate items. Helsabeck (1973) evaluated several hypotheses concerning problem difficulty. In line with premise misinterpretation theorizing, he altered the wording of syllogisms, once to achieve non-ambiguous wording (changing "All A are B" to "Every A is B") and once to achieve a spatial wording of premises (A is inside B, B and C overlap). The former change had no effect and the latter only a small one. Thus, earlier supposi- tions that the problem with logical statements was simply a matter of the meaning of the terms, as used in logic, were not confirmed. Helsabeck then tried three training procedures--spatia1, verbal concrete and verbal abstract--in a refutation task in which introductory students were asked to generate counter-examples to conclusions. The spatial training, utilizing Euler diagrams, facilitated performance the most, with the verbal concrete method of substituting common nouns for alphabetic terms also (though less) facilitative. Verbal abstract training--basically asking §s to verbally work through the logical re- lations of a problem--was not successful in increasing individual's performance level. Helsabeck also found that negative conclusions were harder to refute than affirmative conclusions, but he only used inde- terminate syllogisms in the experiment. Frase (1966) used 1 1/2 hours of programmed instruction, empha- sizing the distinction between formal validity and material truthhood with meaningful items. He also noted differences between problems with either universal or particular conclusions (rather than premises, as used here). His finding was that the training effect he observed was restricted to those problems with particular conclusions, which had, on a pretest, been the more difficult ones. Much training research has used spatial representations as a means for improving performance. DeSoto §t_al, (1965) argue from their data that such a representation acts as a mediator in the solution of three-term series problems. Similarly, in syllogistic research, the results of Helsabeck (1973) and Henle and Michael (1956) argue for the importance of the concept. Schwartz (1971), working with "whb-done-it" problems, suggests three major heuristic elements of a "mode of repre- sentation": (1) it clearly defines needed information; (2) it suggests fruitful orders of operation; and (3) it provides consistency checks. The representation also ought to be generative or productive by defining the problem in such a way that further manipulations may be easily car- ried out. These matters will be brought up again in the discussion section when considering interpretations for the treatment effect. Although a systematic treatment of problem difficulty is lack- ing, the differences between problems has often been noted and attributed to various factors. Frase (1966, 1968b) obtained evidence that the quantity dimension is important, for universals are much easier than particular problems. Lack of clarity regarding the actual problems used, however, urges the caution that the determinate-indeterminate dimension may be involved here, for almost all of the universal problems used in this line of research have been determinate and particulars are, by necessity, indeterminate. Frase (1968a) also investigated the importance of problem figure and found a marginal effect, with figures three and four being more troublesome. He suggests an associative explanation, with the first figure being a forward chain, the fourth a backward chain, and the second and third as stimulus equivalence and response equivalence, respectively. Again, however, it is not certain that the determinateness factor has been properly controlled. For the purposes of the present study, this supposed figure effect may be divided into two possible effects: (1) a difference in difficulty between two problems with the same problem status, but differing in figure (e.g., AA-l and AA-3 are both DUA); (2) a difference in difficulty between two problems identical only in the quantity and quality dimensions (e.g., AE-l is IUM, AE-2 is DUM). The pilot work for this experiment used three types of training. The spatial group practiced on transforming the problems into Euler diagrams, depicting, for instance, "No X are Y" as completely disjoint sets in space. The verbal group was instructed on how to transform the terms of the syllogism into more meaningful sentences, as in "Some ani- mals are not dogs", with the hope that some of the structure of the relations would become clearer. The logical group was encouraged to try the more abstract strategy of thinking of possible cases, in terms of sets and subsets, and attempting to determine a conclusion without using spatial representations or verbal substitutions. A control group was given the same problems to practice on, but with no training instruc- tions; then all of the groups were given an identical test. The results lent but little support to the hypothesis that this "strategy training" would lead to better performance on the test: spatial, logical and control §§ all performed at or near the 55% level, while the verbal §s lagged behind at 49%. A second manipulation in that experiment dealt with the problem status. The finding, consistent across groups, was that the determinate problems were easier than the indeterminate ones; the universals were easier than the particulars, which were, in turn, easier than the mixeds; and the affirmatives were easier than the negatives, with the mixed problems again inferior to all. In addition, specific problems could be noted in terms of these three dimensions and hence, their difficulty could be marked. The easiest problem was a DUA at 90% and the most dif- ficult one, the IMM, was correctly responded to at the 20% level. Apart from the conclusions it suggested concerning problem status, the pilot work underlined the importance of developing training proce- dures that will, in principle (i.e., if followed), be effective enough to produce a solution to a problem. This is necessary in order to dis- tinguish the problems that §s will have in using a procedure from the problems (i.e., of consistency) of the procedure itself. For these reasons, the verbal and logical groups were dropped in the present ex- periment, and an algorithmic one was used to replace them. The question of a training procedure's "completeness" will be considered briefly in the next section. METHOD A. Subjects Subjects (Ss) were 42 (21 male; 21 female) introductory psy- chology students at Michigan State University who had neither had a for- mal course in logic or could demonstrate any reasonable acquaintance with logical methods or principles. §s volunteered for the study and were given credit for their participation. All S; were run individually in the experiment, and were assigned randomly to the treatment conditions. 8. Procedure As experimental §s arrived, they were handed a two-page booklet illustrating the training they were to use (see Appendix A). They were told to read the booklet and encouraged to ask questions about it. They were then shown one of 16 trainingproblems typed on 4" by 6" index cards and told that their task was to find the correct answer from the five alternatives given, concerning what would follow from the top premises. S; were instructed to study the problem, then give the strongest answer possible, and explain their answer. After their answer and their ex— planation, they were told, experimenter (E) and §_wou1d discuss the problem. The list of 16 training problems appear in Appendix B. Scratch paper and a pencil were available, and §s were again asked if they had any questions before they started. 10 ll The two experimental groups differed only in the booklet they received (and were allowed to retain with them during the training ses- sion) and the type of explanations offerred by both §_and S, The spatial group, as shown in Appendix A, were given examples of the application of Euler diagrams to the solution of problems. The first four examples simply show how to diagram the basic four logical propositions. The latter four examples deal with conversion, and demonstrate which prob- lems can or cannot be validly converted. For the algorithmic group, the first page of their booklet consisted of a description of the structure of the problems, including how to decide if a conclusion is valid and how to label each premise. The second page of their booklet shows a tree diagram, illustrating a straightforward, "semi-algorithmic" method of solving the problems. The tree diagram was explained to 5s who could not understand it. The procedure during the training session, as stated above, was , to let S; respond and explain their response before intervening. If either their answer or their explanation was incorrect, it was discussed and the major topics of discussion were as follows: (1) elimination-- §s were encouraged to eliminate hypotheses and to narrow down their prob- lem to a couple of alternatives. This heuristic is explicit in the al- gorithmic condition; in fact, the procedure that was suggested for these §s was to follow the tree as far as they could and then deal with the remaining alternatives (if more than one) by going back to the problem and considering it in terms of sets and subsets (basically, the logical method of the pilot experiment). For the spatial group, the concept of elimination was closely associated with the heuristics of counter-example 12 and refutation, to be discussed below. (2) 00unter-example--§s were encouraged to work negatively, especially if they had narrowed down their hypotheses to two or three., If,_for instance, they were enterr taining hypothesis E, S; were asked "Under what conditions would this hypothesis be false?" and if none, to select it. If all of the hypotheses could be falsified, the indeterminate response would be appropriate and §s were told this if uncertain. (3) Case-conclusion distinction--§s were informed of the distinction between a single case in which the A con- clusion may be true versus the general conclusion that it mu§t_be true. This distinction seemed facilitated by the spatial representation of the separate cases, and indeed, spatial §s were encouraged to number the cases in terms of compatible conclusions, after sustematically diagram— ming all possible cases. In this context, their task was to discover what conclusion, if any, is common to all cases. (4) Refutation--A special instance of counter-example, based on this fact: that conclu- sion A entails both not E and not 0 and that conclusion E entails both not A and not I. It follows from this that any syllogism in which one finds that 99th_alternatives A and E are possible (i.e., possible cases) will be a syllogism in which ng_determinate solution is necessary, be- cause the necessity of each conclusion is refuted by the possibility of both A and E. S; were given a brief explanation of this argument, with emphasis put upon the usefulness of the principle: if one can find cases in which A and E are possible for a given problem, then the prob- lem is indeterminate. (5) Logical sense of "some"--essentially, S; were instructed that the logical propositions I and 0 do not necessarily entail each other. 13 These various heuristics could not be completely standardized across Es for several reasons, including the fact that some subjects make fewer errors than others. Yet all were included in basic form for all of the experimental Es, and certain orders dominated. Refutation followed case-conclusion, and elimination usually preceded counter- example, but often the exact timing of the heuristics was heavily de- pendent upon opportunity and a "sense of the appropriate". Any lack of experimental control that resulted from this flexibility of timing was weighed against the desire for a relatively informal, not overly stress- ful situation for the individual E, During the training session, E kept notes of §fs progress, and of the order in which the heuristics were delivered. Following the 16 training problems, Es were asked to read over their booklets to make sure there wasn't anything they still didn't understand. After they handed back their booklets, the test session began. Es were instructed in the test session as follows. A pack of 32 test problems, face down, was presented to the Es and they were told that this task was the same one as before, to decide which of five prob- lem answers was correct. This time, however, there would be no feedback concerning the answers to the problems by E, nor any explanation, DY.§ or E, of the problems. The §_was then told that he would be given 90 seconds to do each problem, but was admonished that this time period was more than sufficient for most problems and most individuals so that there was no need to hurry. (The time limit was included in the design because preliminary work indicated that, beyond two or three minutes, any extra time given to the subject was of no great benefit. The extra 14 time served only to fatigue the individual and, in fact, most §s worked best when they were working quite rapidly.) Again, scratch paper and a pencil were made available, and the Es were given time to ask questions before beginning. §s simply vocalized their answers to E, who recorded them, and were permitted to work at their own pace. The test session usually lasted between twenty and thirty minutes. The control §s who did not receive training were given the same instructions as the experimental groups for the test session, excluding, of course, any mention of the differences between the training and test sessions. The instructions emphasized that an answer must necessarily follow from the premises to be the correct answer and that one and only one of the five answers is the correct one. The 32 test problems are included in Appencix C. A final aspect of how the experiment was designed deals with the selection of problems for use in the experiment. Several criteria deter- mined the inclusion of both training and test problems, and these will be enumerated in descending importance: (1) determinate items are rare in syllogistic logic, so as many as possible were incorporated into the design. A sufficient number are needed so that Es do not develop an indeterminate response bias, but caution was exercised so that the particular determinates employed in the test set did not overlap exceed- ingly with those of the training set. (2) Each one of the 16 premise combinations was included in the first figure in the test set. (3) The remaining 16 problems of the test set were chosen to facilitate figure (AA-l vs. AA-3), order (EA-2 vs. AE-2), and premise combination (IE-l vs. OA-l) comparisons, as will be discussed later. (4) Finally, 15 dimensional comparisons of the problem status (DUM to be compared with IUM, DMM, DUA, etc.) were included in the design, where possible. In addition care was taken to insure that the letter terms of the problem could not be used as a cue to problem type. Since the problems were typed onto index cards, individual randomization of the order in which the problems were received by the Es was insured by a shuffling of the deck. Status Example (in Figure 1) Dominant response UA All A are B; A11 B are c All A are ca UM All A are B; No B are C No A are C UN No A are 8; No B are C None of the above MA All A are B; Some B are C Some A are C MN All A are 8; Some 8 are not C Some A are not C No A are 8; Some 8 are C Some A are not C MN No A are 8; Some 8 are not C None of the above PA Some A are 8; Some 8 are C None of the above PM Some A are 8; Some 8 are not C None of the above PN Some A are not 8; Some B are not C None of the above aThis is empirically determined, as shown in the next section. Figure 2. Dominant response chart. A final note on the structure of the problem set involves the notion of a dominant response (Figure 2), defined as the form in which ‘a response must be, for each problem, if the problem is determinate. Hence, Figure 2 corresponds to the algorithmic group's tree diagram, in which each problem falls into one of two categories: (1) it is either necessarily indeterminate (UN, MN, PA, PM, PN); or (2) it is either indeterminate, or if determinate, of the form of that problem's dominant 16 response (UM, MA, MM). The lone exception is UA, which takes responses A and I, depending on the figure. The argument to be presented is that Es act as if they know the second category of responses as they enter the laboratory, and include the A response to UA as a non-exceptional instance of the second category. C. Hypotheses The hypotheses may be broken down into those concerning treatment differences, problem differences, sex differences, and possible signifi- cant interactions. The hypothesis was advanced that both experimental groups were to be more successful on the test problems than the control group. This was expected to cover both sexes and all problems, but with the expectation that the greatest treatment differences, in favor of the experimental groups, would be found on the indeterminate problems. Dif- ferences related to sex were not expected, but at least one interaction was plausible: that females might perform better on the spatial train- ing, while males would be more successful on the algorithmic training. No other interactions regarding sex, nor an overall sex effect, were anticipated. Regarding the problems or measures effect, the hypotheses were based primarily on the pilot data that showed determinates easier than indeterminates; affirmatives easier than negatives, with mixed problems even more difficult; and universals, particulars and mixed problems in ascending difficulty. It was anticipated that these dimensions of the stimuli would produce the differences between problems, and that other dimensions would be found not to be very pertinent to problem difficulty. 17 Finally, direct comparison of the two experimental groups receives less emphasis here than the comparison of either to the control group. Honoring the distinction between a particular treatment manifested in this experiment and the more general training features of interest, one must be acutely aware that any difference between the two experimental groups might plausibly be attributed to a difference in construction; although there is sufficient reason for thinking that the conditions are constructed well enough to be comparable to a control group, a higher level of construction would be necessary to compare the conditions with each other, in meaningful fashion. Thus, the comparison of the two experimental conditions with the control group will be emphasized most in the results and discussion sections; a comparison of the experimental groups is left to those tests in which the experimental-control differ— ence does not account for the observed effects. RESULTS A. Training Results Table 1 shows the results of both experimental groups, of both sexes, on the 16 training problems. Since individuals received the Table 1. Scores in percentages on each of 16 training problems. Group-Sex 1. 2. 1 i 2 9. 1 £3. Spatial-Male .571 .714 1.000 (.286 .429 .714 .857 1.000 Spatial-Female .429 .571 .429 .714 .714 .571 .714 1.000 Spatia1-T0tal .470 .642 .714 .500 .571 .642 .785 1.000 Algorithmic-M .286 .714 .286 .714 .714 .714 .714 .857 Algorithmic-F .286 .429 .714 .714 1.000 .857 .714 .857 Algorithmic-T .286 .571 .500 .714 .857 .785 .714 .857 TOTAL .378 .606 .606 .606 .714 .714 .750 .929 Group-Sex _9_ l_q 1_l_ _1_2_ _1_3_ 11 _1_E l_§_ Total Spatial-M .571 .857 .571 .857 .857 .714 .714 .429 .697 Spatial-F .714 .571 .714 1.000 .571 1.000 .857 .714 .705 Spatial-T .642 .714 .642 .929 .714 .857 .785 .571 .701 Algorithmic-M .714 .714 .857 .857 1.000 .714 .571 .714 .697 Algorithmic-F .857 1.000 .714 .714 .857 .714 .857 .857 .759 A1gorithmic-T .785 .857 .785 .785 .929 .714 .714 .785 .728 TOTAL .785 .785 .785 .857 .821 .785 .750 .678 .714 problems in individual random orders, the numbered problems refer not to a specific problem, but rather to a serial order. The major finding 18 19 was an increase in problem success, over all groups, to the level of 70% within five problems and a maintenance of that level until a drop in performance over the last few problems. The training scores for the two experimental groups were then analyzed for the correspondence between training and test scores. A correlation coefficient of .556 (df=26, p=.Ol) was obtained, indicating a tendency for those who score either high or low on the training set to do likewise on the test set of problems. 8. Major Findings The test data were analyzed in terms of the three possible rival hypotheses mentioned earlier, possible figure, order and premise combin- ation effects. In order to separate these effects from those associated with problem status, 16 comparisons were used to explore these hypotheses, but with the stipulation that the problem status was not involved in any comparisons (capturing, for example, a "pure" figure effect). Only the first of two possible figure effects, to take that example, that were distinguished in the introduction is of interest here. The 16 comparisons are shown in Table 2 on the next page. Sta- tistical analysis, which in the case here involves wishing to retain the null hypothesis, was not performed but the lack of variance attributable to these factors, with one noteworthy exception, seems evident. The exception is the AA series, in which the first figure is far easier than the third or fourth. All three may be characterized as DUA problems, but the latter two will be henceforth referred to as DUA' problems for two 20 Table 2. Figure, order and premise combination comparisons total number correct per group (n=l4) per problem. Figure: 9 comparisons ]_ AA-l AA-3 AA-4 E_ AE-l AE-3 E_ AI-l AI-3 S 14 5 5 10 10 9 7 A 14 4 5 9 8 ll 11 C 13 l 2 9 9 14 11 T 41 10 12 28 27 34 29 g_ EA-l EA-2 E_ EE-l EE-2 E_ IA-3 IA-4 S 14 12 12 13 8 10 A 13 10 l4 l4 9 13 C 12 14 5 6 l3 9 T 39 36 31 33 30 32 2_ 11-1 11-2 §_ 00-1 00-2 2_ AE-2 AE-4 S 13 12 12 14 ll 13 A 10 l3 14 13 10 10 C 0 3 3 7 13 12 T 23 28 29 34 34 35 Premise combination: 2 comparisons ]_ EI-l A0-2 0A-3 E_ IE-l OA—l AO-l S 7 4 5 10 11 11 A 10 10 9 12 8 6 C 5 8 10 4 l l T 22 22 24 26 20 18 Order 5 comparisons 1_ AE-2 EA-2 g_ IA-3 AI-3 E_ OA-l AO-l S 11 12 8 7 11 11 A 10 10 9 ll 8 6 C l3 14 13 11 l l T 34 36 30 29 20 18 fl. EO-l OE-l E_ 01-1 10-1 S 8 12 12 14 A ll 12 9 12 C 5 4 2 2 T 24 28 23 28 21 Table 3. Total percentage scores of each group on each problem. Group-Sex DUA DUA' IUA DUM IUM IUN DMA Spatial-Male 1.000 .357 .857 .929 .714 .857 .535 Spatial-Female 1 000 .357 .571 .857 .714 .929 .579 Spatial-Total 1 000 .357 .714 .893 .714 .893 .608 Algorithmic M 1.000 .286 .571 .821 .714 1 000 .857 Algorithmic F 1 000 .357 .429 .714 .500 1.000 .714 Algorithmic T 1.000 .322 .500 .758 .508 1 000 .786 Control-M .857 .000 .143 .929 .000 .285 .893 Control-F 1.000 .071 .143 .893 .286 .500 .786 Control-T .929 .035 .143 .911 .143 .393 .840 Male Total .952 .214 .524 .903 .475 .714 .752 Female Total 1 000 .252 .381 .821 .500 .810 .725 Grand Total .975 .238 .452 .857 .488 .752 .744 Group-Sex IMA DMM IMM IMN IPA IPM IPN TOTAL Spatial-M .543 .429 .557 .714 1.000 .929 .857 .723 Spatial-F .857 .333 .857 .714 .785 .929 .000 .745 Spatial-T .750 .381 .752 .714 .893 .929 .929 .735 Algorithmic-M .571 .519 .557 .857 .929 .543 .000 .755 Algorithmic-f .357 .752 .571 .714 .714 .857 .929 .587 Algorithmic-T .454 .590 .519 .786 .821 .750 .955 .721 Control-M .071 .475 .047 .214 .000 .071 .357 .375 Control-F .143 .519 .238 .429 .214 .214 .357 .455 Control-T .107 .548 .143 .286 .107 .131 .357 .420 Male Total .428 .508 .450 .595 .543 .540 .738 .518 Female Total .452 .571 .555 .519 .571 .557 .752 .532 Grand Total .440 .540 .508 .507 .507 .503 .750 .525 22 reasons: (1) that the results indicate a unique difference between the problems on the basis of figure; and (2) that these DUA' problems require the I response, while AA-l requires the A response. As remarked earlier, the AA problem is unique in this respect, and the empirical determina- tion of the A response as the dominant response of this problem rests on the previously cited fact: a problem with an A response is far easier than those that require the I response. The test data matrix for the design is shown in Table 3. An examination of the data indicates great differences between problems as well as an apparent superiority of the experimental groups over the con- trol group. The data are further displayed in Table 4, which presents the group means on the three dimensions of the problem status. Table 4. Total percentage scores of each group on each problem dimen- sion. Group-Sex Deter. Indet. Univ. Mix. Part. Eff, U15: Egg, Total Spatial-M .633 .794 .786 .582 .929 .667 .735 .809 .723 Spatia1-F .633 .833 .750 .673 .905 .691 .734 .881 .746 Spatial-T .633 .814 .768 .628 .917 .679 .735 .845 .735 Algorithmic-M .724 .777 .738 .724 .857 .714 .704 .952 .755 Algorithmic-F .694 .682 .667 .643 .833 .595 .684 .881 .687 A1gorithmic-T .709 .730 .703 .684 .845 .655 .694 .917 .721 Control-M .684 .121 .441 .408 .135 .393 .384 .286 .375 'Control-F .694 .286 .536 .490 .262 .429 .510 .429 .465 Control-T .689 .204 .489 .449 .199 .411 .447 .358 .420 Tota1-M .680 .564 .655 .571 .640 .591 .608 .682 .618 Total-F .674 .600 .651 .602 .667 .572 .643 .730 .632 Grand Total .677 .583 .653 .587 .659 .582 .626 .706 .626 23 Because the instances of problems used in the experiment were not equally represented (for reasons given in the methods section), analysis of the data was made by means of the percentage of correct re- sponse, per problem and per individual. Since this mode equates all problems in the analysis, an analysis was also performed on the total scores for each individual (i.e., with the problems differentially weighted, as they were in the experiment). The former data set will be referred to as the "original data", and the latter as the "total scores data". In addition, to account for the differential veriability of problems, a "transformed data" set was created by dividing each individ- ual's score on each problem by the standard deviation of the problem. Analyses of variance were then run on each of these data sets, using conservative tests with an alpha level of .05. Table 5 shows the repeated-measures analysis of variance for the original data. The sources of variation of significance are the treat- ment effect, the measures or problems effect, and the treatment-measures interaction. All three exceeded the .05 level of significance. Table 6 shows the transformed data, indicating the same three significant results. Table 7, ignoring the effect of problems, confirmed the treat- ment effect. C. Treatment and Treatment-measures Effects In order to more precisely determine the limits of the differences between treatments, 14 univariate analyses of variance were performed. The results of these tests and the Scheffé contrast confidence intervals (Scheffé, 1959) resulting from them are enumerated in Table 8. 24 Table 5. Analysis of variance using original data. Source E 9i: _M_S E Treatment (T) 18.045 2 9.023150 35.85a Sex (S) .044 1 .043940 0.18 TS .767 2 .383567 1.57 I:TS 8.816 36 .244893 ---- Measures (M) 20.070 13 1.543822 18.55a 12.857 25 .494898 5.94a SM .744 13 .057223 0.69 TSM 1.615 26 .062108 0.75 IMzTS 38.995 468 .083322 ---- Table 6. Analysis of variance using transformed data. Source ES_ d_f E E Treatment 202.481 2 101.24052 40.15a Sex .794 1 .79449 0.32 TS 9.115 2 4.55726 1.81 I:TS 90.780 36 2.52166 ---- Measures 1305.520 13 100.50923 113.84: TM 172.719 26 6.64303 7.52 SM 9.708 13 .74673 0.85 TSM 17.457 26 .67141 0.76 IMzTS 413.202 468 .88291 ---- Table 7. Analysis of variance using total score data. Source EE g: M_S_ E Treatment 866,175 2 433,087 21.45a Sex 1,093 1 1,093 0.05 TS 51,765 2 25,882 1.28 I:TS 726,885 36 20,191 ---- ap < .05 25 Table 8. Fourteen two-way ANOVAs and Scheffé contrast confidence intervals. Problem Source F value Upper limit Lower limit DUA Treatment (T) 1.0000 .681 -.397 Sex (S) 1.0000 TS 1.0000 DUA' T 3.3182 1.432 -.218 S 0.1818 TS 0.0455 IUA T 5.4444 1.861 -.005 S 1.0000 TS 0.3333 DUM T 2.6308 .420 -.742 S 1.6615 TS 0.1385 IUM T 8.9178* 1.881 .189 S 0.0411 TS 1.5205 IUN T 26.4643* 1.773 .441 5 1.7143 TS 0.7500 DMA T 2.5120 .448 -1.020 S 0.1627 TS 1.0301 IMA T 10.7647* 1.832 .168 S 0.0441 TS 1.2353 DMM T 3.3735 .744 -.798 S 0.4243 TS 0.6611 IMM T 22.2432* 1.791 .399 S 1.4395 1.4402 IMN T 6.5821* 1.757 .099 S 0.0448 TS 0.8507 26 Table 8. Continued. Problem Source F value Upper limit Lower limit IPA T 38.0571* 2.204 .796 S 0.7714 TS 3.0857 IPM T 30.6923 2.141 .693 S 1.9231 TS 0.5385 IPN T 25.5938 1.869 .491 S 0.0938 TS 0.6563 *Indicates an F value significant at .05/14=.0036. All contrasts were run as S + A - 2C. A11 significant contrasts favor S + A. The results are relatively clear-cut. Not only do the S and TS effects not appear in the overall analysis, but they are similarly absent from each two-way analysis reported in Table 8. The treatment-measures interaction can be formulated in the following manner: the indetermin- ates, with the exception of IUA, all show a treatment effect (all favor- ing the experimental groups), while the determinates show no treatment effect. It is noteworthy that DUA' of all determinates, comes closest to exhibiting a treatment effect. Figure 3 shows the results graphi- cally. Two two-way analyses of variance were then run, which summarize these 14 tests. Again, the determinate problems show no training effect (Table 9), but the indeterminate problems manifest one well beyond the critical F ratio for significance at .05 (TablelO). Taken together 28 Table 9. Analysis of variance using determinate problems. Source df SS MS F Treatment 2 43,456 21,728 .7546 Sex 1 378 378 .0131 TS 2 3,122 1,561 .0541 I:TS 36 1,036,597 28,794 ---- Table 10. Analysis of variance using indeterminate problems. Source df SS MS F Treatment 2 3,089,150 1,544,580 52.810 Sex 1 22,791 22,791 .925 T5 2 101,515 50,807 2.055 I:TS 35 885,202 24,589 ----- with the fourteen earlier analyses, these results implicate that the control group does as well as the experimental groups on the determinate problems, but is far inferior on the indeterminates. 0n those problems that are necessarily indeterminate (negative or particular problems), the control group inferiority is extended. 0n those problems that are usually or often determinate (universal, affirmative, or mixed in either way), the margin of difference is smaller. Figure 4 illustrates these dimensional comparisons. The analysis, however, may go a bit further. As mentioned earlier, some of the indeterminate problems are indeterminate by virtue of either a negative value on the quality dimension or a particular value on the quantity dimension. These indeterminates shall be 29 1.001- — SPATIAL r‘ ”f“ ALGORITHMIC .50 _ - CONTROL l I DETERMINATE INDETERMINATE SPATIAL - ALGORITHMIC .50 _ ' ‘ ' 1 9.4925544: _ UNIVERSAL MIXED ALGORITHMIC _- / SPATIAL .. 0—7 - .___ .50 — _.__ ’ VT? *—-0CONTROL 0 b 1 I L AFFIRMATIVE MIXED NEGATIVE Figure 4. Per cent correct response, per group, on three dimensions of problems. 30 designated "Group A" (including IUM, IMN, IPA, IPM, and IPN). "Group B" indeterminates are not so easily characterized, and a hypothetical psychological correlate of a figure change is responsible for their recognition as indeterminates. What is clearer is that they are excep- tions to general rules, those rules specified by the dominant responses of each problem status. A brief look at Table 8 or Figure 3 demonstrates that the dif- ference between experimental and control groups--favoring the experi- mental groups in all cases--is somewhat larger in the Group A than in Group B (IMM, IMA, IUM, IUA). This difference may be due to the follow- ing consideration: for Group A problems, falsification (i.e., selecting the "none of the above" response) is a one-step operation. For Group 8 problems, one has to (1) refute the dominant response of that problem (Figure 2) and then (2) falsify the other three determinate solutions. For the following test, the hypothesis was that control group Es would do poorer on Group 8 problems than experimental Es because of the first step, not the second step. The hypothesis was operationalized in the following manner. Nine problems were selected, on the basis of their being easily categorized into "rules" or "exceptions". A rule was defined as a problem that had a dominant response which was the correct response for that problem. This is the set of determinate problems, excluding DUA'. An exception was defined as a problem which had a dominant response, but one in which that response was not the correct one. This is the set of Group B indeterminates. DUA' was added to the list of exceptions because of the 31 Rules ExcepEions DUA IUA, DUA' DUM IUM DMA IMA DMM IMM Figure 5. Rules and exceptions. manner in which the group was defined. It differs from the others in that its correct response, though also not the dominant response, is not the indeterminate response, either, but rather another determinate response. It is argued that although the problem is structurally a determinate problem, it functions as an exception. Four categories of response were discriminated: (1) see the rule--marking a determinate response to any of the four rules; (2) see the correct rule--marking the correct response, given that one has marked a determinate response; (3) see the exception--marking a non-dominant response to any of the five exceptions; and (4) see the correct excep- tion--marking the correct response, given that one has marked a non- dominant response. These four types of responses were recorded for each individual on all four sets of problems. The summaries, over all prob- lems, are given in Table 11; the same data, analyzed for each problem, are in Appendix 0. Four two-way analyses of variance were performed on the following data (Tables 12-15). The analyses indicate no treatment differences on the four analyses. Scheffé tests were used to examine the three remain- ing significant treatment effects and revealed a superiority of the two experimental groups over the control group (S+A-20) on "see exception“, 32 Table 11. Total percentage scores of each group on each category. See Correct See Correct Group—Sex Rule Rule Exception Exception Spatial-M .857 .792 .786 .783 Spatial-F .762 .907 .815 .859 Spatial-T .810 .846 .800 .821 Algorithmic-M .798 1.000 .786 .762 Algorithmic-F .786 .955 .586 .807 Algorithmic-T .792 .978 .687 .782 Control-M .833 .957 .186 .308 Control—F .868 .918 .357 .520 Control-T .853 .937 .271 .448 Male Total .830 .915 .587 .723 Female Total .805 .926 .587 .772 Grand Total .817 .920 .587 .748 but on "correct exception" the experimental-control difference was slightly less significant, barely missing the .05 level. Because the control group performed at a higher level than the spatial group on the "correct rule", two-way contrasts were employed here. No difference were found between either the control group and the algorithmic group or the control group and the spatial group; however, the algorithmic group was superior to the spatial group on this category. In terms of the hypothesis stated earlier, the results do indi- cate that control Es perseverate longer on the dominant response than the experimental groups, but it is also shown that even when they are able to see an exception, the proportion of times that the control group are able to see the correct exception is also inferior to that of the ex- perimental groups (though it statistically eludes the .05 level). The Table 12. Analysis of variance for "see rule". Source SS df MS F Treatment (T) 3.76190 2 1.88095 1.22 Sex (5) .85714 1 85714 0.56 TS 4.42858 2 2.21429 1.43 I:TS 55.42857 36 1.53968 ---- Table 13. Analysis of variance for "correct rule". Source SS df MS F T 93,890,439 2 45,945,219 4.09a 5 28,133 1 28,133 0.00 TS 45,746,719 2 22,873,359 1.99 I:TS 412,999,670 36 11,472,213 ---- Table 14. Analysis of variance for "correct exception". Source SS df MS F T 11,088,522 2 5,544,311 12.35a S 435,337 1 435,337 0.97 TS 206,182 2 103,091 0.23 I:TS 16,165,181 36 449,032 ---- Table 15. Analysis of variance for "see exception". Source SS df MS F T 215.57143 2 108.28572 25.55a S 0 1 0 0.00 TS 24.57143 2 12.28572 2.91 I:TS 152.00000 36 4.22222 ---- ap < .05 34 lack of a "see rule" effect indicates no treatment differences in the selection of determinate solutions to determinate problems. Figure 6 summarizes the differences between experimental and control groups quite neatly: (l) the control group is inferior on the Group A indeterminates and (2) the control group is inferior on the Group B indeterminates, which have been broken down into the constituent abilities of "see exception" and "correct exception". On both of these categories, the control group was found to be inferior to the experi- mental groups. 0. Measures Effect The significant measures effect shown in Tables 5 and 6 was scrutinized by dimensional comparisons in each of the three types of problems. Figures 7 and 8, on succeeding pages, illustrate the compari- sons to be investigated. The rationale for the tests that were performed deserves some explicit recognition. The hypotheses entertained at the outset of the experiment dealt with the three dimensions of the problem status; the post-hoc analysis of the problem set into Determinate, Group A and Group 8 problems enlarges this conception of the possible Measures effect and, at the same time, affords a more convincing and meaningful method of testing the original hypotheses. The plan is as follows: tests were run on the dimensional com- parisons shown in Figures 7 and 8, but only ijEjfifeach of the three problem types. This leaves seven ANOVAs--one between the three prob- 1em types and two within each problem type--and they are more stringent 35 .mEmFroa 40 mqup muggy co .azocm emu .mmcoammc pumscoo ucmu com .o mesmwm A.<=o wzHozsquv A.<=o wzHeamuxmv m azomw < 830mm me 3. Some X are Y (B) (C) 111 This statement has three possible diagrams. Any of these cases may (A) illustrate the relationship between X and Y. 55 4. Some X are not Y L (A) (B) (C) O :0 Again, there is more than one figure possible for the single statement. Since we do not know which is the correct figure, we must consider all of them. Now look at these examples: 5. All X are Y Therefore, All Y are X. Is this a valid conclusion? To decide this, ou look at (A) both possibilities (the ones in #1 . The statement 0R "All Y are X" is true for B, but not for A. $0, the conclusion is not valid. (B) 6. No X are Y Therefore,- No Y are X. Is this a valid conclusion? Again, as in #2, we have only one diagram possible. The answer is yes. .1 LY . ‘ ’ OR (Em) 0R . (A) ” (a) 8. 57 Some X are Y Therefore, . Some Y are X. Is this a.va1id conclusion? Look at the three pos- sibilities, and notice that the statement is true for all three. The answer is yes. (C) Some X are not Y Therefore, Some Y are not X. Is this a valid conclusion?* Note the second possibility here. The answer is no. 0R (A) 58 FORM A This is an experiment in logical syllogisms. A syllogism con- tains two premises and one conclusion, of the form below. Premise 1 (true) Premise 1 (true) Premise 2 (true) Premise 2 (true) Conclusion (true)--Valid conclusion Conclusion (false)--invalid conclusion In this experiment, we are assuming the truth of the two premises, and you are asked to decide if any of a number of conclusions is also true; if it is, then it is called a valid conclusion; if it is not, it is an invalid conclusion. "True" means true in gygpy_case; for a conclusion to be valid, it must be true whenever the two premises are. The statements that form the premises and conclusions are illus- trated below in numbers 1-4. 1. A11 X are Y.--Affirmative Universal This statement is called affirmative because it is in positive form (it says what all X are, not what they aren't). It is called universal because it refers to all X5. 2. No X are Y.—-Negative Universal This statement is called negative because it tells us what the X5 aren't, not what they are. It is universal because all Xs are in- cluded: All of them are not Y. 3. Some X are Y.--Affirmative Particular This is affirmative because it is in positive form, and particu- lar because it is referring only to some of the X5. 59 4. Some X are not Y.--Negative Particular. This is negative because it tells us what some of the X5 are not; it is particular because it refers to only some of the $5. Affirmative Negative Universal 1 2 Particular 3 4 60 _* :owmapucou pmmgm>wcz e>eeaeeeee< LO .a>eeaeewee< m* :owmzpucou empzupuceq LO .mw commzpucou eepa> oz :opmz—ucou uppm> oz mm” gmpzuv ewe» ea m* :owmapucou empaowucma m>_pmscpmm< sf «cmpzuwucma Ems» Lo mco ma non mg¢ / \ ew>pummoc Ems» we «so mH oz cowmz m* Pucou. evens oz cowmafiucou. Pemcm>wcz Na a>eeamez oz wsnpaowusma swsu.mo mco mm V. wgmpzowugma swzu we even mc< \\\\\\ mm> ~m>wpmamc mmmwsmeq span ms< L . o copmzpucou .1 uwpm> oz commspocou L mm cmpauwpcmn o cowm:_ucou ¢* m>pummmz vam> ozme mm> cowmzpucou m* twpo> oz mm>\ cowmapucou m* vam> oz mm> APPENDIX B TRAINING ITEMS Premises AE EA AE AE Figure 4 Status DUM DUM DUM IUM IUM 66 Correct 2 Problem All J are K No K are L Therefore, 1. A11 L are J 2. No L are J 3. Some L are J 4. Some L are not J 5. None of the above No P are 0 ' All R are P Therefore, 1. All R are 0 2. No R are 0 3. Some R are 0 4. Some R are not Q 5. None of the above No T are U All V are U Therefore, 1. All V are T 2. No V are T 3. Some V are T 4. Some V are not T 5. None of the above All M are N No 0 are N Therefore, 1. All 0 are N 2. No 0 are N 3. Some 0 are N 4. Some 0 are not N 5. None of the above All S are T No S are U Therefore, 1. All U are T 2. No U are T 3. Some U are T 4. Some U are not T 5. None of the above Premises EE E1 E0 IA IE Figure 3 Status IUN DMM IMN IMA IMM 62 Correct 5 Problem No D are E No D are F Therefore, 1. A11 F are E 2. No F are E 3. Some F are E 4. Some F are not E 5. None of the above No J are K Some J are L Therefore, 1. All L are K 2. No L are K 3. Some L are K 4. Some L are not K 5. None of the above No T are U Some V are not U Therefore, 1. All V are T 2. No V are T 3. Some V are T 4. Some V are not T 5. None of the above Some A are B All C are 8 Therefore, 1. All C are A 2. No C are A 3. Some C are A 4. Some C are not A 5. None of the above Some Y are X No Z are X Therefore, 1. All 2 are Y 2. No Z are Y 3. Some 2 are Y 4. Some 2 are not Y 5. None of the above Premises EE EE AI AI IA Figure 1 Status IUN IUN DMA DMA DMA 67 Correct 5 Problem No D are E No F are 0 Therefore, 1. All F are E 2. No F are E 3. Some F are E 4. Some F are not E 5. None of the above No M are N No 0 are N Therefore, 1. All 0 are M 2. No 0 are M 3. Some 0 are M 4. Some 0 are not M 5. None of the above All T are V Some U are T Therefore, 1. All U are V 2. No U are V 3. Some U are V 4. Some U are not V 5. None of the above All 0 are M Some 0 are N Therefore, 1. All N are M 2. No N are M 3. Some N are M 4. Some N are not M 5. None of the above Some A are 8 All A are C Therefore, 1. All C are 8 2. No C are 8 3. Some C are 8 4. Some C are not B 5. None of the above Premises IA IA AI EI A0 Figure 4 Status DMA IMA IMA DMM DMM 68 Correct 3 Problem Some Y are X All X are Z Therefore, 1. All Z are Y 2. No 2 are Y 3. Some Z are Y 4. Some Z are not Y 5. None of the above Some U are W All V are U Therefore, 1. All V are W 2. No V are W 3. Some V are W 4. Some V are not W 5. None of the above All A are 8 Some C are 8 Therefore, 1. All C are A 2. No C are A 3. Some C are A 4. Some C are not A 5. None of the above No G are H Some F are G Therefore, 1. All F are H 2. No F are H 3. Some F are H 4. Some F are not H 5. None of the above All H are F Some G are not F Therefore, 1. A11 G are H 2. No G are H 3. Some G are H 4. Some G are not H 5. None of the above APPENDIX C 'TEST ITEMS Premises AA AA AA AE Figure 1 TEST ITEMS Status DUA DUA' DUA' IUA DUM Correct 65 1 Problem All A are 8 All C are A Therefore, 1. All C are B 2. No C are 8 3. Some C are 8 4. Some C are not B 5. None of the above All P are 0 All P are R Therefore, 1. All R are 0 2. No R are 0 3. Some R are 0 4. Some R are not Q 5. None of the above All F are G All G are H Therefore, 1. All H are F 2. No H are F 3. Some H are F 4. Some H are not F 5. None of the above All Y are X All Z are X Therefore, 1. All Z are Y 2. No Z are Y 3. Some Z are Y 4. Some Z are not Y 5. None of the above All E are F No D are F Therefore, 1. A11 0 are E 2. No D are E 3. Some 0 are E 4. Some 0 are not E 5. None of the above Premises OE II II 10 OI Figure 1 Status IMN IPA IPA IPM IPM 70 Correct 5 Problem Some V are not W No X are V Therefore, 1. All X are W 2. No X are W 3. Some X are W 4. Some X are not W 5. None of the above Some H are I Some J are H Therefore, 1. All J are I 2. No J are I 3. Some J are I 4. Some J are not I 5. None of the above Some J are K Some L are K Therefore, 1. All L are J 2. No L are J 3. Some L are J 4. Some L are not J 5. None of the above Some R are S Some T are not R Therefore, 1. All T are S 2. N0 T are S 3. Some T are S 4. Some T are not S 5. None of the above Some N are not 0 Some P are N Therefore, 1. All P are 0 2. No P are 0 3. Some P are 0 4. Some P are not 0 5. None of the above Premises OO 00 Figure 1 Status IPN IPN 71 Correct 5 Problem Some C are not 0 Some E are not C Therefore, 1. All E are D 2. No E are D 3. Some E are D 4. Some E are not 0 5. None of the above Some Y are not W Some X are not W Therefore, 1. All X are Y 2. No X are Y 3. Some X are Y 4. Some X are not Y 5. None of the above APPENDIX D RULE-EXCEPTION RESULTS PER PROBLEM Group Spatial Algorithmic Control Total Group Spatial Algorithmic Control Total Group Spatial Algorithmic Control Total Group Spatial Algorithmic Control Total RULE-EXCEPTION RESULTS Problem UA Rule Correct Rule See dddd . . C . Problem UM Rule Correct Rule See PER PROBLEM Exception .714 .594 .190 .500 Exception Correct Exception .667 .640 .500 .635 Correct Exception Problem MA Rule Correct Rule See .786 .714 .143 .548 Exception .909 .850 1.000 .891 Correct Exception Problem MM Rule Correct Rule See .857 .643 .143 .583 Exception .792 .722 .429 .714 Correct Exception .881 .714 .452 .683 .865 .867 .316 .744