Two models of syllogistic reasoning: Feature selection and conversion

JOURNAL OF VERBAL LEARNING AND VERBAL BEHAVIOR

14, 180-195 (1975)

Two Models of Syllogistic Reasoning: Feature Selection and Conversion RUSSELL REVLIS1 California State Universtty

The present paper offers two processing models of how reasoners solve categorical syllogisms. The models are based on traditional statements of the atmosphere effect and the conversion hypothesis. A test of the two models shows that previous studies of formal reasoning have unnecessarily restricted the scope of the hypotheses and have failed to compare them on the critical condmons and in their intended senses. Both models are reasonably accurate in predicting the overall distribution of errors. While the feature selection model is superior to the conversion model m predicting the decisions on a set of critical problems, the underlying assumption of the feature selection model is not supported by the data. The formal, or categorical, syllogism has made a useful contribution to our knowledge of inferential processes by providing a welldefined task environment in which such processes may be probed (cf. reviews by Henle, 1962; Johnson, 1972; Wason & Johnson-Laird, 1972). The syllogism consists of a pair of premises followed by one or more conclusions: All M are P Some S are M Therefore: (1) All S are P (2) No S are P (3) Some S are P (4) Some S are not P (5) None of the above

(Major premise) (Minor premise) (Conclusions)

The reasoner's task is to judge whether the relation holding between the subject and predicate terms of the conclusion (S and P) can be unambiguously inferred from the relations expressed in the premises. The formal rules for making this judgment are expressed in their traditional forms by Cohen and Nagel (1934) and Peirce (1957) and are

summarized here in the Appendix? Formal syllogisms have found their way onto (a) general intelligence tests (e.g., Thurstone, 1938), (b) clinical protocols (e.g., Gottesman & Chapman, 1960; Von Domarus, 1944), (c) evaluations of attitudes and prejudices (e.g., Gordon, 1953; Janis & Frick, 1943; Kaufman & Goldstein, 1967; Morgan & Morton, 1944; Winthrop, 1946), (d) assessments of belief systems (e.g., McGuire, 1960a, b, c), and (e) investigations of memory processes (e.g., Erickson, 1972; Frase, 1966; Whimbey & Ryan, 1969). Despite the widespread use of syllogisms, strong theorizing on the fundamental processes entailed in such reasoning has only recently been forthcoming (cf. Erickson, 1974). In this context, the present paper offers two models of syllogistic reasoning and provides a test of their empirical predictions. Historically, only two major hypotheses have been proposed to deal with formal reasoning; and these are offered to account for only a portion of the errors that reasoners

2 A contrasting modal logic analysis of syllogisms has been offered by Rescher (1969), which he claims 11 express my apprematlon to S. Havfland, N. Revlis, to be closer to Aristotle's own approach to categorical L. Rips, E. Smith, and the anonymous revxewers for reasoning, Partial support for the psychological reality their thoughtful comments on earher versions of this of such modal systems is offered by Revlis and Hayes (1972) in a different reasonmg task. paper. Copyright © 1975 by Academic Press, Inc. 180 A l l rights o f reproduction m any form reserved Printed in Great Britain

SYLLOGISTICREASONING make. These hypotheses are known as the atmosphere effect (Sells, 1936; Sells & Koob, 1937; Woodworth & Sells, 1935) and the conversionhypothesis (Chapman & Chapman, 1959). 3 The atmosphere effect claims that reasoners' deductions are based on a featurematching scheme and that subjects do not attend to the meaning of the propositions they are asked to reason about. In contrast, the conversion hypothesis claims that errors are due to a miscoding of the propositions and not to faulty inference. Tests of these hypotheses have proven inconclusive (e.g., Begg & Denny, 1969; Simpson & Johnson, 1966; Stratton, 1967). There are at least three critical reasons why current research has failed to provide a major test of the conflicting explanations for syllogistic reasoning. First, the current hypotheses (they could hardly be called theories) are so vaguely specified that no strong test of them may even be possible in their present form. Second, the tests that have been performed have been confined largely to a class of problems where the reasoner cannot draw a logically valid conclusion from the information presented. Certainly, any model offered to account for formal reasoning must be required to make predictions for the entire gamut of problems--those with a valid conclusion as well as those with no valid conclusion. Third, because of ambiguities in the initial specification of the hypotheses, research has failed to compare them on the critical conditions and in their intended senses. With these considerations in mind, the present paper undertakes to recast the two hypotheses as processing models of syllogistic reasoning and to provide a first test of them. T w o MODELS OF REASONING Feature Selection The primary emphasis of the research with 3An alternative hypothesis, based on transfer paradigms has been suggested by Frase (1966, 1968) and by Pezzoli and Frase (1968) The hypothesis has not had a major impact on reasoning research.

181

syllogisms has been an assessment of the reasoner's inference mechanisms. Such studies examine both the logical and alogical operations in formal reasoning. The major thrust o f this research has been directed at testing a single hypothesis, the atmosphere effect. This hypothesis was first proposed by Woodworth and Sells (1935) to account for the data o f Wilkins (1928). It claims that errors in formal reasoning are due to the intrusion of an alogical process: The reasoner judges the vahdity of the conclusions to syllogisms based on a match between a "global impression" formed from the premises with one formed from the conclusion (cf. a restatement of the hypothesis by Begg & Denny, 1969). That is, when the reasoner does not comprehend the relationship between the premises and the conclusion offered for his inspection, his judgment will be influenced by the atmosphere (global impression) of the premises. Although this hypothesis was formulated to account for reasoning on indeterminate syllogisms (for which no vahd conclusion is possible), its principles can be extended to deal with reasoning on valid syllogisms as well: A reasoner following the atmosphere principles will be correct on 7 9 ~ of the syllogisms with valid conclusions since these principles are direct analogs of the validity rules for categorical judgments (Cohen & Nagel, 1934). The applicability of the atmosphere hypothesis to reasoning on valid syllogisms has been only briefly explored (e.g., Wason & Johnson-Laird, 1972); 4 the results (though inconclusive) generally disconfirm the hypothesis. Tests of the hypothesis on invalid syllogisms have been equivocal: Its confirmation has been supported by some (e.g., Begg & Denny, 1969; Simpson & Johnson, 1966; 4 Wason and Johnson-Lalrd (1972) presented pilot data on a small number of valid syllogisms. However, more than a third of the problems did not correspond to the structural requirements determining the order of terms and the major and minor premises (these rules have peen disregarded by others: e g., Begg & Denny, 1969; Ceraso & Provitera, 1971; Sells, 1936).

182

RUSSELL REVLIS

Stratton, 1967) and challenged by others (e.g., Ceraso & Provitera, 1971; Chapman & Chapman, 1959; Henle, 1962; Wason & Johnson-Laird, 1972). There are at least two strong criticisms of the atmosphere hypothesis. First, the hypothesis neither embodies a model of reasoning nor explains the presence of errors in syllogistic reasoning. That is, the hypothesis does not specify why reasoners should fail to grasp the nature of the relationshlp between subject and predicate terms, nor does it provide a statement of the overall probability of an error for any specific problem. The atmosphere effect is a contingent statement of the preponderant error on invalid syllogisms, given the existence of any error: It is a data description on a limited set of problems; it is not a model of decision making in formal reasoning. The second criticism is that the predictive accuracy of the hypothesis may be an artifact of the constraints of the task environment, including the instructions and the composition of the problems. The instructions frequently direct the reasoner to accept an invalid conclusion. For example, in one study (Begg & Denny, 1969) subjects were instructed to put a question mark in the margin of the booklet if they could not find a conclusion which validly followed from the premises. Such manipulations may pressure reasoners into accepting one of the propositional concluslons. In addition, subjects are frequently presented with a preponderant number of invalid syllogisms (often 70 ~ of the problems have no valid conclusion). This factor may enhance the likelihood of an invalid conclusion being accepted. As the Chapmans (1959) indicate, subjects do not expect that most of the problems they will encounter cannot be answered. If they feel compelled to make some selection other than "None are proven" or a question mark in the margin, the atmosphere principles might seem a reasonable basis for making their validity judgments. In this case, the atmosphere effect

would decidedly reflect the constraints of the task rather than indicate that reasoners do not possess formal inference mechanisms. It should be added that the hypothesis cannot be completely dismissed on this basis since atmosphere-like errors also appear in noncategorical reasoning tasks (Evans, 1972a, b, 1973a, b; Hunter, 1957, 1958). If the essential principles underlying the atmosphere hypothesis are to have any value for explaining the validity judgments in formal reasoning, the vagueness of the hypothesis must be eliminated and the hypothesis recast as a model of reasoning. The following is a reformulation of the atmosphere effect as a feature selection model. It is described as a series of processes which permit us to specify the reasoner's representation of the propositions as well as the operations he performs on that representation when making validity judgments. This feature selection model describes the solution to formal syllogisms in terms of four stages, as shown in Figure 1. 'READ EXTRACT."

I /

Features J

(Prernlse-P1) 1

1

READ EXTRACT Features (Premise- P2)

yes

JENTERE3: < LCornl: oslte •

oo

,

/~PPLY' 1 RU_E-2

yes

J

ENTER I"i j~ Composlterp

I

D J EXTRACT: ~ Featurer~ I -

JREA (Co.~1.'cn)l

n=n÷1

[

--

yeu

FIG. 1. F e a t u r e s e l e c t i o n m o d e l o f f o r m a l r e a s o n i n g .

SYLLOGISTIC REASONING

Stage 1: Premise representation. In the first stage, the reasoner forms a single representation of each premise in the syllogism. This representation is based on an extraction of two features from the premises: quantity and polarity. The quantity attribute refers to whether or not a statement is universally quantified; its values are expressed as [_+Universal]. The polarity attribute refers to whether or not a statement is affirmative; its values are expressed as [+Affirmative]. Stage 2: Composite representation. The composite representation (CMP) is formed in two steps and is guided by two rules. Rule 1: If the two premises have the same sign on an attribute, the composite sign for that attribute is the sign of any single premise for that attribute. Rule 2: If the signs differ for an attribute, the sign of the composite representation for that attribute is minus (i.e., particular or negative). In the first step of this stage, the reasoner compares the values for the first attribute [_+Universal]. Rule 1 is presupposed in the coding of the composite representation unless there is a mismatch on the attribute values, in which case Rule 2 is applied. Figure 1 shows that when a mismatch occurs, one additional operation is required to form the composite representation (i.e., Rule 2 must be applied). The second step proceeds in an identical manner as the first for the analysis of the second attribute [_+Affirmative]. Table 1 illustrates the outcome of stages one and two for all premise-pairs (for clarification of the notation, see the appendix).

Stage 3: Representation of the conclusion. When a conclusion (Cn) is considered by the problem solver, he extracts the relevant features of the proposition and represents it using the operations described for Stage 1. Stage 4: Comparison process. In this stage, the composite representation is compared with that of the conclusion ( C M P : C , ) - - a t present, speculations concerning the serial or parallel nature of such a comparison would be premature. If the composite representation and the conclusion are congruent,

183

TABLE 1 FEATURE SELECTION REPRESENTATION

Premise: Quantity: Polarity:

[Quantity,polarity] [+ Universal] [+ Affirmative]

Types: A: E: I: O:

All A are B [+, +] No A are B [+, -] Some A are B [-, +] Some A are not B [-, -] Composite Premise-pair representation AA: [+, +1 & [+, +] [+, +1 AE: [+, +] & [+, -] [+, -I AI: [+, +] & [-, +] [-, +] AO: [+, +] & [-, -] [-, -1 EI: [+,-] & [-, +1 [-, -1 EO: [+,-] & [ - , - ] [-, -] EE' [+,-] & [+,-] [+,-] n: [-, +1 & [-, +1 [-,+1 Io: [-, +] & [-, -1 [-, -1 oo: [-,-1 & [-,-1 [-,-1 the reasoner accepts the conclusion. If the two representations are incongruent, the reasoner either responds "invalid" or, as a function of the paradigm, reads the next conclusion (On+l). Predictions. This formulation of the atmosphere hypothesis as a feature selection model has at least one major advantage over the previous descriptions: The modelis sufficiently detailed to make predictions concerning the solutions to every syllogism. The traditional description made a single prediction: The dominant error, when solving invalid syllogisms, will be to accept the conclusion that has the same atmosphere (attribute-values) as the premise-pair. The present model not only makes the same predmtion, it goes farther and offers a series of testable claims concerning the solution process. The predictions of the model fall into two categories: (a) It specifies the relative error rates of all syllogisms under modest time constraints and (b) it offers a qualitative estimate of its own accuracy--the conditions under which the reasoner's decisions will deviate from the model. The model makes the following three claims

184

RUSSELL REVLIS

concerning the reasoner's error rates. First, the subject will make a logically incorrect decision on all invalid syllogisms. For these problems, the logically correct decision would be to respond "none"; however, this response cannot be made since it does not correspond to any possible set of features in Stage 2. The second claim is that the subject's decisions will be logically correct on approximately 79% of the valid syllogisms. The logically correct conclusion corresponds to the composite features on 15 problems (AA-1, AO-2, OA-3, AE-2, AE-4, EA-1, EA-2, AI-1, AI-3, IA-3, IA-4, EI-1, EI-2, EI-3, EI-4), but not on the remaining four valid syllogisms (AA-3, AA-4, EA-3, EA-4). Third, it follows from the first two claims that (a) for 21% of the valid syllogisms--those where the model prescribes a logically incorrect decision--error rates should be equal to that for the invalid syllogisms since the same erroneous decision rules operate on both of these valid and invalid problems; (b) performance on valid syllogisms (on the average) will be better than on invalid syllogisms (optimally, 21% errors for valid syllogisms, 100 % errors for invalid ones). Deviations from the model's prescribed responses can be predicted if we make the strong assumption that each required operation is error prone and, therefore, the likelihood of the subject deviating from the model will increase with the number of component processes necessary to make his response. The major source of deviation will result from Stage 2, where the number of required operations varies with the application of Rule 2. That Is, the use of Rule 2 requires at least three operations while the use of Rule 1 requires only two. Simply stated, the accuracy of the feature selection model in predicting the decisions will be inversely related (in a weak sense) to the number of operations required to establish the composite representation in State 2. This assumption of operational complexity offers three additional implications for the

subject's reasoning errors. First, for valid syllogisms where the model prescribes a logically correct decision (correct valids), deviations from the model should result in an increase in error rate. Second, for valid syllogisms where the model prescribes a logically incorrect response (incorrect valids), deviations from the model should result In a decrease in error rate. Third, the number of operations in Stage 2 will not affect the performance on invalid syllogisms since no deviations in the composite representation will ever correspond to the logically correct conclusion, "none." Conversion of Propositions

The conversion hypothesis is based on the view that errors result primarily from an incorrect encoding of the propositions and not from the operation of a faulty inference mechanism. The major source of misinterpretation is said to be illicit conversion. That is, when the reasoner is told that All A are B, he may interpret this proposition to mean that the converse, All B are A, is also true. Illicit conversion as a source of errors in syllogistic reasoning was suggested by Chapman and Chapman (1959), although it was also noted by Wilkins (1928). An example of how conversion might operate is illustrated with the following two syllogisms: (2) All M are P (1) All P are M Some M are S Some S are M Therefore: Therefore: Some S are P No conclusion is valid In syllogism (1) we cannot completely determine the relation between S and P from the information provided in the premises. If a student were reasoning logically on this problem, he would claim that no valid conclusions were possible. However, if while encoding the premises, the reasoner converts each one in turn, the problem, under conversion, would appear to him as syllogism (2). This converted syllogism does have a solution,

185

SYLLOG1STIC REASONING

Some S are P. Therefore, if the reasoner converts the propositions, a new problem is produced with a conclusion that is inappropriate for the original syllogism. The Chapmans (1959) claim that conversion is based on two factors: first, on our experience of reality and, second, on the ambiguity of the copula is a. With respect to the first factor, we may empirically (though not logically) accept the converse of many O (see Appendix) propositions: Some plants are not green may be converted to an empirically true proposition, Some green things are not plants. Consequently, when reasoning about abstract material, the reasoner may feel justified in assuming that Some A are not B is equivalent to S o m e B are not A. The second factor claims that there is a tendency to encode the copula is a as is equal to, rather than the logical is included in. Recent linguistic discussion of "conversational implications" have provided empirical support for the two bases of conversion in normal speech (Gordon & Lakoff, 1971; Horn, 1972; Lakoff, t970--see also the category-interchange hypothesis of Meyer, 1970). Despite evidence for the process of conversion in normal speech, the hypothesis as originally proposed leaves considerable gaps in its specification of the reasoning process. Left unclarified are: (a) the necessary and sufficient conditions for conversion, (b) whether one or both of the premises are converted, and (c) where in the reasoning process such conversions occur. Neither the Chapmans (1959), nor any of the researchers examining their hypothesis, have considered these insufficiencies. The following is a reformulation of the conversion hypothesis cast as a process model. Because of obscurities in its initial presentation by the Chapmans (1959), several models might be drawn directly from the stated hypothesis. Consequently, a single version of the conversion model must suffice to illustrate the general class of models. This version is shown in Figure 2.

/bEDUCE\ ~,Composlte

,5

c°~d~'°°'C~

l

y~B

@ FI~. 2. Conversion model of formal reasoning.

Three strong assumptions are made in the conversion model. First, it is assumed that for syllogisms with abstract propositions, the conversion operation is obligatorily present in the initial encoding of all propositions. Clearly, this represents the strongest interpretation of the conversion hypothesis. The second assumption is that the reasoner is "biased" against accepting a non-propositional conclusion (e.g., a "none" response). When faced with accepting such a conclusion, the reasoner will either re-work the problem (time permitting) or make a guess from among the propositional conclusions. This assumption is supported by the foregoing discussion of response bias in tests of the atmosphere hypothesis. Third, when making repeated passes through the problem, the reasoner is assumed to restrict his interpretation of the propositions so that (either consciously or mechanically) the conversion interpretation is blocked. With these strong assumptions in mind, the following is a statement of the model and the predictions which are drawn from it. Stage 1: Encoding. In the first stage, the reasoner reads-in the premises, applies a conversion operator in the process of e n -

186

RUSSELLREVLIS

coding, and stores a representation for the individual propositions. A notational form for this representation may be a bracketed proposition or a pictorial scheme such as Venn (or Euler) diagrams (e.g., Erickson, 1974; Johnson-Laird, 1970; but also see Pylyshyn, 1973). As yet, no complete notational form has been successfully presented. Stage 2: Composite representation. The composite representation (CMP) of the premise-pair is the result of an unspecified deduction operation (either immediate or mediate deduction). The output of this stage is a single proposition relating the subject and predicate terms of the conclusion. Again, this may be in the form of a deep structure sentence (proposition) or Venn diagram. The notational form is necessarily constrained by that given for Stage 1. Stage 3: Conclusion encoding. The conclusion (C,) is presumed to be encoded with conversion and represented in a form amenable to a comparison with the output of Stage 2. Stage 4: Comparison. The reasoner compares the output of Stage 2 with the conclusion encoded in Stage 3. If the two propositions are congruent, the reasoner responds "valid." If the two are incongruent, the reasoner considers the next conclusion offered. For those problems where none of the propositional conclusions match the reasoner's composite predicate (i.e., a "none" conclusion is the only one remaining for him), the reasoner tries to either work the problem again (make a second PASS through the problem) or make a GUESS from among the alternatives. The PASS mechanism is initiated by a "none" conclusion and is limited by the time available (PASS LIMIT). The reasoner, reevaluates his knowledge of the syllogism by making a second pass through it--restricting his interpretation of the syllogism's premises. This PASS mechanism may consist of a more cautious re-reading of the displayed propositions or a recovery of the initial propositions by a back-up operation in memory. If, in the

latter case, the conversion interpretation is viewed as the last in a series of derived interpretations to be given the premises before the subject works the problem, the "back-up" operation accesses a meaning closer to the one that is logically presupposed (i.e., a less derived interpretation). If the additional PASS does not result in a propositional conclusion being appropriate, the reasoner makes a GUESS from among the available conclusions. For the present, this may be taken to be a "fair" guess. However, an alternative view considered later is that the feature selection model may serve as the GUESS mechanism. Predictions. The traditional conversion hypothesis claimed that reasoners make logical deductions on all syllogisms. When errors occur, they are the result of having ilhcitly converted the premises so that the original syllogism is transformed into one with a different required conclusion. The conversion model described here makes the stronger claim that conversion is intrinsic to the representation of the premises on all abstract syllogisms, so that the reasoner never makes deductions from the same propositions that are provided him by the experimenter. The predictions from the model are described separately for invalid and valid syllogisms. The model distinguishes between two types of invalid syllogisms: (a) those where conversion transforms the syllogism into one which has a different propositional conclusion than the problem presented by the experimenter, called DIFFERENTS, and (b) those where conversion produces a syllogism with the same conclusion as the one presented by the experimenter, where "none" is the correct decision, called NONES. On the first PASS through all syllogisms, the representation of the premises entails a conversion operation. For DIFFERENTS, the reasoner will accept the propositional conclusion required by his representation and his decision will be scored as "incorrect." For NONES, the converted syllogism reqmres a nonpropositional, "none"

SYLLOGSITIC REASONING

conclusion; on these problems, the reasoner will make a second PASS through the premises and re-represent them in an unconverted form. However, this will again result in a nonpropositional, "none" conclusion. In this case, the reasoner will make a (fair) GUESS from among the possible conclusions. As a result, correct "none" responses will occur on approximately 20 ~ of the problems (when there are five alternatives to choose among). The workings of the PASS and GUESS mechanisms produce an overall conversion function for invalid syllogisms: ideally, 80 errors for NONES, 100~o errors for DIFFERENTS. The conversion model distinguishes among three types of valid syllogisms: (a) those where conversion results in the same conclusion as the presented problem, SAMES, (b) those where conversion produces a syllogism with a different propositional conclusion than the presented one, DIFFERENTS, and (c) those where conversion results in a nonpropositional conclusion, NONES. For SAMES and DIFFERENTS, the reasoner accepts the conclusion logically required by his representation: SAMES decisions will be scored as "correct" and DIFFERENTS decisions will be scored as "incorrect." However, for NONES, the reasoner makes a second PASS through the problem without a converted interpretation of the premises and accepts the logically appropriate conclusion; his responses will be scored as "correct." Therefore, the reasoner's decisions on SAMES and NONES should show fewest errors (ideally, zero percent error) while DIFFERENTS should show maximum errors (ideally, 100 ~ error). A conversion function should, therefore, be shown for valid syllogisms as well as for invalid syllogisms. Feature Selection and Conversion

In none of the attempts at comparing the conversion and atmosphere hypotheses on invalid syllogisms has there been an unambiguous finding for one hypothesis over

187

the other. However, a recent study by Ceraso and Provitera (1971) showed that by disambiguating the premises, the kind and frequency of errors can be altered. These data are consistent with the notion that errors are the result of misrepresentation of the premises. Unfortunately, in that study, the syllogisms were non-standard (they violated the logical principles governing the ordering of the major and minor premises). In addition, while the results of disambiguation support the miscoding hypothesis, they do not provide unequivocal evidence for the presence of a conversion operator at encoding. In contrast with the traditional forms of the atmosphere and conversion hypotheses, the present specifications of the feature selection and conversion models make strong, testable, and differentiating predictions concerning reasoning performance. With respect to overall error rates, the feature selection model distinguishes two categories of valid syllogisms and a single, homogeneous category ot invalid ones; the conversion model makes predictions for three categories of valid syllogisms and two categories of invalid ones. In addition, both models prescribe the decisions the reasoner should reach on each problem. This permits a comparison of the models since they prescribe different decisions for the reasoner on a set of critical problems (AA-1, AA-2, AA-3, EA-1, EA-2, EA-3, EA-4, AE-2, and AE-4). The experiment presented here is offered as a first test of these models in a task environment which differs in three important ways from previous studies. First, subjects are traditionally limited only in the time they are given to solve an entire set of problems, thereby obscuring between-problem difficulty. Second, valid syllogisms are often poorly sampled and rarely analyzed. Finally, many of the recent studies employ syllogisms that violate one or more of the structural rules of formal syllogisms. The present experiment controls the time per problem, controls the proportion of valid syllogisms,

188

RUSSELLREVLIS

and is consistent with the rules of formal syllogisms.

METHOD Students were asked to solve 64 formal syllogisms where letters represented category names. Half of these problems had a valid conclusion and half did not. This ratio represents a departure from traditional procedures which employ a preponderance of invalid syllogisms. There were 18 unique valids and 21 unique invalids with repetitions providing the full set of problems. The syllogisms were selected to ensure a broad coverage of mood and figure (see Appendix). The valid syllogisms were: AA-1, AA-3, AA-4, AI-1, AI-3, IA-3, IA-4, AE-2, AE-4, EA-1, EA-2, EA-4, AO-2, OA-3, EI-1, EI-2, EI-3, EI-4. The invalid syllogisms were: AA-2, AI-2, AI-4, IA-1, IA-2, AE-1, AE-3, AO-1, OA-1, II-1, II-2, IO-1, IO-2, OI-1, OI-2, IE-1, IE-2, EE-1, EE-2, OO-1, 00-2. The ordering of the problems within the booklets was randomized with the restriction that runs of more than three instances of a single problem or validity-type were not permitted. Procedure

The subjects were randomly assigned to two groups (n = 25) which differed only on the amount of time permitted to solve each syllogism, 15 or 30 seconds. The students were told that they would be required to solve reasoning problems where their goal was to decide which of five possible conclusions (All C are A, No C are A, Some C are A, Some C are not A, and None o f the above is proven) had to follow from two given premises. The subjects read the definitions of the four categorical propositions (A, E, I, and O) and worked a sample problem (AI-3). The experimenter provided the correct answer to the practice syllogism and explained why other possible decisions were incorrect.

The students were instructed to work each problem in the time allotted and to proceed to the next problem in their booklets only when told to do so. The instruction to "turn the page" was tape-recorded and presented over headsets. Subjects

The reasoners were 50 men and women fulfilling a course requirement for introductory psychology. None of the subjects had been exposed to a course in logic. They were run in groups of four, in sessions lasting between 20 and 40 minutes. A 2-minute rest period was permitted after the subjects solved the first 32 problems.

RESULTS The performance of each reasoner was determined by summing the errors for each mood-figure combination. An overall percent error score for each problem was then used for analysis. The two time periods allowed for solution (15 and 30 seconds) showed no difference in performance in any analysis. In addition, time did not significantly interact with other factors in the experiment. Although several problems were repeated, no practice effects were observed. Feature Selection

The data are presented in Table 2 and support the model's claim that the error rates in solving valid syllogisms increase with the number of operations in Stage 2, F(2, 144) = 27.9, p < .001. However, the precise distribution of errors only partially supports the model's predictions with respect to operational complexity: (a) Errors on correct valids increased with the number of applications of Rule 2, F(2, 96) = 20.7,p < .001, as predicted; (b) there is no difference in the error rates for incorrect valids as a function of the number of applications of Rule 2, contrary to predictions. These results contribute to an inter-

t 89

SYLLOGISTIC REASONING

action between the number of times Rule 2 is applied and the two kinds of Validity, F(1, 48) --- 5.5, p < .05. The model incorrectly claims that performance on invalid syllogisms will be unaffected by the number of times Rule 2 is required. Table 2 shows that error rates actually increase with the number of steps necessary to form a composite representation for these problems, F(2, 96) = 18.0, p < .001. This is due to a difference in performance between syllogisms which require the exclusive use of Rule 1 and those which require at least one application of Rule 2, F(1, 9 6 ) = 8.7, p < .01. There is no difference in performance as a result of which attribute, quantity or polarity, Rule 2 operates upon, or how many times it operates. TABLE 2 FEATURE SELECTION MODEL: PERCENT ERROR AND PERCENT OF THE DECISIONS CORRECTLY PREDICTEDa

Number of apphcatmns of Rule 2 Problem-type

0

1

cribed response for each problem with the response actually made by each subject. Table 2 summarizes the model's accuracy for the various experimental conditions and shows that the feature selection model correctly predicts 7 1 . 4 ~ of the subjects' decisions. This accuracy should decrease as the number of operations required in Stage 2 increase, if the formation of a composite representation proceeds in a manner claimed by the model. This change in the predictive accuracy does occur for invalid syllogisms, F(1, 96) = 8.8, p < .01, and for correct valids, F ( 2 , 9 6 ) = 7 1 . 7 , p < . 0 0 1 , but not for incorrect vahds. Overall, the feature selection model predicts correct valids with greater accuracy than incorrect vahds, matched for the number of operations in Stage 2, F(1, 48) = 13.5, p < .001. The model's accuracy on incorrect valids does not differ from invalid syllogisms (matched for the number of operations in Stage 2). Conversion Model

2

The data were partitioned according to the conversion model's specifications and are Correct valids 4.0(93.0) 10.5(88.3) 23.9(61.6) presented in Table 3. The results support the Incorrect valids 91.0(82.4) 84 0(72.0) -Invahds 79.2(67.8) 90.2(72.4) 92.8(58.4) TotaP 58.1(81.1) 61.6(77.6) 58.4(60.0) TABLE 3 "The pre&ctive accuracy is given an parentheses. CONVERSION MODEL: PERCENT ERROR AND PERCENT The totals are based on unwelghted means. O17 THE DECISIONS CORRECTLY PREDICTEDa The overall distribution of errors conforms to the model's distinction among the different problems. First, the subjects make fewer errors on correct valids than incorrect valids (matched for the number of operations in Stage 2), F(1, 48) = 499.2, p < .001. Second, the error rates for decisions on incorrect valids are equivalent to that found for invalid syllogisms (matched for the number of operations in Stage 2). Third, the present data support previous findings that subjects make fewer errors on valid syllogisms than on invalid ones, F(1, 48) = 902.7, p < .001. The predictive accuracy of the model was determined by comparing the model's pres-

Vahdlty

SAMESb

NONES DIFFERENTS

Vahds Invalids Total

27 2(19.1) -27.2(19 1)

28 4(69.7) 88 0(88 0) 84 7(--)c 93.5(63.0) 56.6(69.7) 91.2(75.5)

"The pre&ctaveaccuracy Is given in parentheses. b SAMES includes problems whose concluslons under conversmn are either Identical to the conclusions for the presented problem or a subset of them. c The predictave accuracy is undefined for these problems. model's claim that syllogisms which have different conclusions in the converted form have higher error rates than those whose conclusions are unaffected by conversion,

190

RUSSELLREVLIS

F(1, 48) = 23.9, p < .001. This conversion effect holds both for syllogisms with a valid conclusion, F(1, 48) = 22.3, p < .001, and for those with no valid conclusion, F(1, 48)= 14.5, p < .001. The effects of the response bias are discussed separately for valid and invalid syllogisms. Invalid syllogisms. When the encoding of invalid syllogisms produces a syllogism with a propositional conclusion (DIFFERENTS), the reasoner will accept this conclusion and his decision will be scored as "incorrect." When the converted syllogism logically requires a nonpropositional conclusion (NONES), subsequent reinterpretations of the premises without conversion (PASS) do not alter the conclusion. The model predicts, therefore, that the reasoner makes a GUESS from among the five alternative conclusions and he will make the correct decision on 20 of the problems. The data support these predictions: (a) The reasoners make an "incorrect" decision on D I F F E R E N T invalid problems (error rate --93.5~); (b) the reasoners are correct 15.3 ~ of the time on problems requiring a guessed response (NONES), which compares favorably with the predicted 2 0 ~ correct decisions. Valid syllogisms. There are three types of valid syllogisms describedpreviously: SAMES, NONES, and DIFFERENTS. The conversion model predicts that the error rates on SAMES and NONES should be equivalent since the reasoner will make the correct deduction on both problems (for NONES, this will occur on the second pass through the syllogism). In addition, both SAMES and NONES should have a lower error rate than DIFFERENTS since for the latter problems, the reasoner readily accepts an incorrect propositional conclusion. The data show that there is an overall difference in observed error rates for the three types of valid syllogisms, F(2, 96) = 303.2,p < .001. SAME and NONE problems do not differ in their error rates

(27.2 and 28.4 ~, respectively), while both are clearly lower in error than D I F F E R E N T problems (88.0 ~ errors). The conversion model further claims that the error rates on valid DIFFERENTS should be equivalent to that for invalid DIFFERENTS, since, for these problems, the reasoning mechanisms are identical. The results show that the two types of DIFFERENTS are equivalent in their error rates (88.0~ and 93.5~ for valid and invalid problems, respectively). The accuracy of the conversion model is summarized in Table 3 (excluding invalid NONES), which shows that the model correctly predicts 61.0~ of the reasoner's decisions (chance = 20~). For valid syllogisms, the accuracy is higher for DIFFERENTS (88.0~) than for SAMES s and NONES averaged together 44.7 ~, F(1, 98) = 108.4, p < .001. For invalid syllogisms, the model correctly predicts 63.0 ~ of the decisions on DIFFERENT& For the remaining invalid syllogisms, the model is correct in claiming that the overall performance is at chance levels (chance = 20 ~, observed = 15.3 ~o).

Two Models Compared Previous comparisons of the atmosphere and conversion hypotheses have been largely restricted to invalid syllogisms and primarily to AO-3 problems (e.g., Simpson & Johnson, 1966). However, in their present formulations, the feature selection and conversion models do not differ in the responses they predict for the AO-3 problems: Under both models, the reasoner is said to respond with an O conclusion. In contrast with the previous hypotheses, the present models make differentiating predictions on a broader range of problems. In the present selection of problems, the two models may be contrasted on 5 This category includes problems where either of two decisions are logically correct. The model ~s accurate in claiming that the reasoner will make a logically valid choice; it has difficulty, however, m specifyingthe precise decisionreached.

SYLLOGISTICREASONING eight valid syllogisms (AA-1, AA-3, EA-1, EA-2, EA-3, EA-4, AE-2, and AE-4) and a single invalid syllogism (AA-2). Before testing the two models, it is necessary to distinguish two kinds of valid syllogisms: (a) unitary problems, where each model makes a single prediction which conflicts with the alternative model's prediction (e.g., for AA-2 problems, feature selection predicts an A conclusion, conversion predicts that an I conclusion will be accepted); (b) dual problems, where the conversion model is affirmed if either of two decisions are made, while the feature selection model makes a single prediction. For example, if All S are P is an appropriate conclusion for a syllogism, Some S are P is also logically appropriate. The conversion model predicts that either of these two responses will be made, while the feature selection model predicts only one decision. On dual problems, the two models are neither mutually exclusive (as with the unitary problems), nor are they identical in their predictions. A comparison of the two models on the unitary problems shows that the feature selection model is overwhelmingly the better predictor of the reasoner's decisions. For the valid syllogisms (AA-1, AA-3, EA-1, and EA-2), the feature model accounts for 88.9 ~o of all decisions, while the conversion model accurately predicts only 6.8 ~o of the decisions. On each of these problems, both a universally quantified and a particularly quantified proposition are logically appropriate. In every case, the feature selection model prescribes the universal and the conversion model prescribes the particular. For the invalid syllogism (AA-2), the feature selection model is correct on 85.2~ of all decisions, while conversion is correct on 8.4 ~ of the decisions. In contrast, there is little difference between the models on the dual problems (EA-3, EA-4, AE-2, and AE-4): feature selection yields 79.4~; conversion, 85.7~. Overall, the two processing models jointly account for 91.9 ~ of all decisions on the critical problems described here. 7**

191 DISCUSSION

I

There are two critical components of the feature selection model: (a) an alogical acceptance of conclusions based on a featurematching operation and (b) an information processing hypothesis that the more operations entailed in making a decision, the greater the deviation from the ideal predictions of the model. Solely on the basis of the primary, feature-matching operation, the model's predictive accuracy averages 71.4Yo over all problems in this study. In contrast, the claims of the complexity hypothesis achieve only modest support from the data. However, deviations from the predictions need not be critical, since they are based on a considerably underspecified mechanism, For example, the reasoner was assumed to misrecall the features of quantity and polarity as having different attributevalues, but these new values were never specified, and the time allotted may not have been sufficiently brief to observe the interaction of time and processing complexity. The present formulation of the conversion model is successful in accounting for the traditionally observed effect of validity as well as indicating regularities in the data which have hitherto gone unnoticed. It correctly distinguishes among traditionally equated problems and predicts that there is a conversion function relating the reasoner's errors to the availability of a different conclusion in a converted syllogism. The predictive accuracy of the model is close to that of the feature selection model (61.0 ~ compared with 71.4~--for those problems where the models are unambiguously specified, discounting the complexity mechanism of the feature selection model and the guessing mechanism of the conversion model). In contrast, the conversion model appears to be incapable of predicting the precise response that a reasoner will make on five of the problems which critically distinguish between the conversion model and the feature

192

RUSSELL REVLIS

selection model. However, the conversion model should not be dismissed solely on this basis. While the feature selection model is more accurate in its predictions, its motivating princlple appears to be contradicted by the data: If subjects are making their decisions based on feature-matching and not on logical inference, the model's accuracy should be uniform across conditions (controlling for complexity). Yet, the model is most accurate when its prescribed decision corresponds to the logically correct one (correct valids, 90.7 ~ ; incorrect valids, 77.2 ~ ; matched for complexity). In addition, the model is incorrect in predicting uniform error rates across invalid syllogisms because reasoners make logically correct decisions for NONES. Together these facts suggest that the feature selection model fails to account for the observed rationality of the subjects and that its higher predictive accuracy may be spurious. Interestingly, the accuracy of the conversion model is lowest on those syllogisms where the reasoner makes a logically correct decision, but not the specific logically correct decision prescribed by the model. Consequently, while the model's predictive accuracy is lower than the feature model the data do support the motivating principle of the conversion model --namely, that the reasoner's decisions are guided by rational processes. The most unsatisfactory component of the conversion model is the GUESS mechanism, which functions as a free parameter. Future work on such a model will require a more formal specification of this mechanism; a strong candidate for it is the feature selection model. That is, feature selection may be a subprogram invoked when logical inferences fail to result in a propositional conclusion. For those problems where conversion predicts a fair guess, the observed decisions were decidedly biased toward the feature selection predictions. The propositional conclusions matched the features of the premises on 60.8 ~o of the decisions. If the conversion model is amended so that when the GUESS mechanism

is invoked, guesses of propositional conclusions are biased towards feature selection, the predictive accuracy of the conversion model on invalid syllogisms will be 76.1 on guessed decisions and 71.8 ~ overall. This view of the role of feature selection seems more appropriate than Sells' (1936) claim that atmosphere decisions occur when the subject fails to see the relationship between the subject and predicate terms of the syllogism. In the present framework, the reasoner's use of feature-matching is a result of a response bias against accepting nonpropositional conclusions. The reality of such a bias is suggested by the data of Moore (1974), who showed that when reasoners are provided with an estimate of the percentage of invalid syllogisms, they show an improvement in performance on only those invalid syllogisms where the GUESS mechanism would be invoked--no changes are shown for other invalid syllogisms or for any of the valid syllogisms. Analogous results are provided by Henle and Michael (1956), who informed their reasoners that they were to solve syllogisms which had a valid (i.e., propositional) conclusion. The reasoners were consistent in n o t responding according to the predictions of the feature selection model. This supports the notion that feature selection may be a last resort, when inference fails. The present description of the feature selection and conversion models represents the first occasion on which the atmosphere and conversion hypotheses have been realized as processing models of formal reasoning-and their predictive accuracies assessed. The present experiment is offered as a first test of the two models. The present study differs from previously published reports in (a) the explicitness of the predictions, (b) the control over the problem environment, including the amount of time the reasoners had to work each problem as well as the proportion of valid and invalid syllogisms, and (c) the range of problems analyzed. The present concern has been to determine whether the models

SYLLOGISTICREASONING can serve as a basis for future research and theorizing. In this sense, both models are viable, but both models are insufficiently specified. The feature selection model, in its present form, is underspecified if the complexity hypothesis is to be taken seriously. Without the additional mechanism, differences between sub-classes of problems would go unexplained. The high predictive accuracy of this model clearly argues for its consideration, although the data contradict its underlying assumption of an alogical reasoner. 6 While the feature selection model is sufficiently specified to be actualized on a computer, the conversion model is n o t - owing, in part, to inexplicitness of the composite representation (Stage 2). This portion of the conversion model embodies the inference mechanism. Although formal rules for such inferences are not likely to be forthcoming (cf. a discussion by Wason & JohnsonLaird, 1972), psychological rules may be derived and are essential if the model is to be seriously offered as an explanation of categorical reasoning. Clearly, the absence of any such mechanism is a fundamental failing of the feature selection model as well. One possible basis for the inference mechanism is a decision stage, whose output is based on the construction and analysis of Venn diagrams. This possibility is close at hand since work is already progressing both on verification of sentences against diagrams (Johnson-Laird, 1970; Neimark, 1975, in press) and on the use of diagrams in reasoning about syllogisms (Ericksen, 1974; Whimbey & Ryan, 1969; see also Ceraso & Provitera, 1971). Alternatively, a propositional framework has been successfully used to model linear syllogisms (e.g., Clark, 1969) and it may prove effective in both specifying varieties of misrepresentation (inversions, obversions, 6 The predlctive accuracy of the atmosphere hypothesis may be related to the kinds of materials used it has been found to be as low as 20 ~ for syllogismswith concrete propositions (Morgan & Morton, 1944).

193

conversions) and in capturing different forms of inference that may occur in categorical reasoning (Henle, 1971). Both models presented here represent specific instances of two classes of models: those which emphasize alogical operations (feature selection) and those which emphasize misrepresentation of the information (conversion). The exemplars of these classes of models may vary with respect to whether specific operations occur in series or in parallel (e.g., feature-extraction and feature-comparison) and whether such operations occur uniformly on all propositions (e.g., conversion may occur obligatorily during the first PASS through the problem, but optionally in subsequent tries). The feature selection and conversion models are offered here as first approximations to formal models of categorical reasoning. They both incorporate the essential features of their original hypotheses, while avoiding much of the vagueness. While these models are based on the traditional hypotheses concerning formal reasoning, they are clearly not restatements of them. It is the nature of such modeling that the products often have properties of their own and they must stand on their own. Hopefully, the construction of these models will serve as a theory-based impetus for future research.

APPENDIX Each proposition in a formal syllogism asserts a relation between two sets of terms. The major premise expresses a relation between the predicate term of the conclusion (P) and another term (M), as in All M are P. The minor premise expresses a relation between the subject of the conclusion (S) and the second term (M), as in Some S are M . Within each premise, the terms may appear in two orders (M--P or P--M, M--S, or S--M). Consequently, there are four possible combinations or orderings for each syllogism.

194

RUSSELL REVLIS

These orderings are called figures:

CLARK, H. H. Linguistic processes in deductive reasoning. Psychological Review, 1969, 76,

Fig. 1 M--P S--M S--P

387-404. COHEN, M. R., & NAGEL,E. An introduction to logic. New York: Harcourt, 1934. ERXC~SON,J. R. Memory search in formal syllogistic reasoning. Paper Presented at the Meeting of the Midwestern Psychological Association, 1972. ERICKSON, J. R. A set analysis theory of behavior in formal syllogistic reasomng tasks. Loyola Symposium on Cognition, 1974, in press. EVANS, J. ST. B. T. Reasoning with negatives. British Journal of Psychology, 1972, 63, 213-219. (a) EVANS,J. ST. B. T. Interpretation and "matching bias" in a reasoning task. Quarterly Journal of ExperimentalPsychology, 1972, 24, 193-199. (b) EVANS, J. ST. B. T. On the problems of interpreting reasoning data: Logical and psychological approaches. Cognition, 1, 373-384, 1973. EVANS,J. ST. B. T., & LYNCH,J. S. Matching bias in the selection task. British Journal of Psychology, 1973, 64, 391-397. FRASE, L. T. Validity judgments of syllogisms in relation to two sets of terms. Journal of Educational Psychology, 1966, 57, 239-244. FRASE,L. T. Associative factors in syllogisticreasoning. Journal of Experimental Psychology, 1968, 76, 407-412. GORDON, R. Attitudes toward Russia on logical reasoning. Journal of Social Psychology, 1953, 37, 103-111. GORDON,D., & LAKOFF,G. Conversationalpostulates. Seventh Regional Meeting of the Chicago Linguistic Society. Chicago: University Press, 1971. GOTTESMAN,L., & CHAP~C~N,L. J. Syllogistlc reasoning errors in schizophrenia. Journal of Consulting Psychology, 1960, 24, 250-255. HENLE,M. On the relation between logic and thinking. Psychological Review, 1962, 69, 366-378. HENLE, M. Of the scholler of nature. Social Research, 1971, 38, 93-107. HENLE, M., & MICHAEL,M. The influence of attitudes on syllogistic reasoning. Journal of Social Psychology, 1956, 44, 115-127. HORN, L. R. On the semantic properties of logical operators in English. Unpublished Ph.D. dissertation, Linguistics Department, University of California, Los Angeles, 1972. HUNTER, I. Note on an atmosphere effect m adult reasoning. Quarterly Journal of Experimental Psychology, 1957, 9, 175-176. HUNTER, I. Skill m human reasoning. Proceedings of the Royal Physical Society of Edinburgh, 1958, 27, 33-39. JANIS, L., & FRICK, F. The relationship between

Fig. 2 P--M S--M S--P

Fig. 3 M--P M--S S--P

Fig. 4 P - - M (Major premise) M - - S (Minor premise) S--P (Conclusion)

The premises in each syllogism are selected from only four types of propositions, determined by the orthogonal pairings of two features: quantification (universal and particular) and polarity (affirmative and negative). These four propositions, and their traditional letter designations, A, E, I, and O, are: A: E: I: O:

AltAareB No A a r e B Some A are B Some A are not B

Since any of these four types of propositions may be the major premise and any of the four types may comprise the minor premise, there are 16 combinations of premise-types, called moods. Each of these sixteen moods can appear in any of the four figures. Consequently, there are 64 possible syllogisms--only 19 of which have a valid conclusion. With this information in mind, the sample problem shown on the first page can be characterized as having an A-type major premise and an I-type minor premise. It has, therefore, an AI mood and is written in the first figure. Its designation is "AI-I". The conclusion that validly follows from the premise is an I proposition.

REFERENCES

BEGG, I., & DENNY, J. Empirical reconcihation of atmosphere and conversion interpretation of syllogistic reasoning. Journal of Experimental Psychology, 1969, 81, 351-354. CERASO, J. t~ PROVITERA, A. Sources of error in syllogistic reasoning. Cognitive Psychology, 1971, 2, 400-410. CHAPMAN,L. J., & CHAPMAN,J. P. Atmosphere effect re-examined. Journal of Experimental Psychology, 1959, 58, 220-226.

SYLLOGISTIC REASONING attitudes toward conclusions and errors in judging logical validity of syllogisms. Journal of ExperimentalPsychology, 1943, 33, 73-77. JOHNSON,D. M. Systematic introduction to the psychology of thinking. New York: Harper & Row, 1972. JOHNSON-LAIRO,P. N. The interpretation of quantified sentences. In Flores D'Arcais & W. J. M. Levelt (Eds.), Advances in psycholinguistics. New York: American Elsevier, 1970. KAtrrMAN,H., & GOLDSTEIN,S. The effects of emotional value of conclusions upon distortions in syllogistic reasoning. Psychonomic Science, 1967, 7, 367-368. LAKOrF, G. Linguistics and natural logic. Synthese, 1970, 22, 151-271. McGumE, W. J. Cognitive consistency and attitude change. Journal of Abnormal and Social Psychology, 1960, 60, 345-353. (a) McGumE, W. J. Direct and indirect persuasive effects of dissonance-producing messages. Journal of Abnormal and Social Psychology, 1960, 60, 354358. (b) McGuIRE, W. J. A syllogistic analysis of cognitive relationships. In M. J. Rosenberg & C. I. Hovland (Eds.), Attitude organization and change. New Haven: Yale University Press, 1960. (c) MEYER, D. E. On the representation and retrieval of stored semantic information. Cognitive Psychology, 1970, 1,242-299. MOORE, W. D. Polarity effects in syllogisticreasoning. Unpublished Masters Thesis, California State University, Fullerton, 1974. MORGAN, J. J., & MORTON, J. T. The distortion of syllogistic reasoning produced by personal convictions. Journal of Social Psychology, 1944, 20, 39-59. NEIMARK, E. Development of comprehension of logical quantifiers. In R. Falmagne (Ed.), Psy-

chological studies of logic and its development. Potomac, MD: Erlbaum, 1975, in press. PEmCE, C. S. The syllogism. In J. M. Baldwin (Ed.), Dictionary of philosophy and psychology, Vol. 2. Gloucester: Peter Smith, 1957. PEZZOLI,J. A., & FRASE,L. T. Mediated facilitatlon of syllogistic reasoning. Journal of Experimental Psychology, 1968, 78, 228-232. PYLYSHYN,Z. W. What the mind's eye tells the mind's brain: A critique of mental imagery. Psychological Bulletin, 1973, 80, 1-24.

195

RESCHER,N. Aristotle's theory of modal syllogisms and its interpretation. In N. Rescher (Ed.), Essays in philosophical analysis. Pittsburgh: University of Pittsburgh Press, 1969. REVLIS,R., • HAYES,J. R. The primacy of generalities in hypothetical reasomng. Cognitive Psychology, 1972, 3, 268-290. SELLS, S. B. The atmosphere effect: An experimental study of reasoning. Archives of Psychology, 1936, 29, 3-72. SELLS,S. G., & KOOB, H. F. A classroom demonstration of "atmosphere effect" in reasoning. Journal of EducationalPsychology, 1937, 28, 514-518. SIMVSON,M. E., & JOHNSON,D. M. Atmosphere and conversion errors in syllogisticreasomng. Journal of ExperimentalPsychology, 1966, 72, 197-200. STRATTON,R. Atmosphere and conversion errors in syllogistic reasoning with contextual material and the effect of differential training. Unpublished masters thesis, Michigan State University, 1967. THURSTONE,L. L. Primary mental abilities. Psychometric Monograph, No. 1. Chicago: University of Chicago Press, 1938. VON DOMARUS,E. The specific laws of logic in schizophrenia. In J. S. Kasanin (Ed.), Language and thought in schizophrenia. Berkeley: University of California Press, 1944. WASON,P. C., & JOHNSON-LAIRD,P. N. Psychology of reasoning: Structure and content. London: Batsford, 1972. WnI~EY, A., & RYAN, S. Role of short-term memory and training in solving reasoning problems mentally. Journal of Educational Psychology, 1969, 60, 361-364. WILKINS, M. C. The effect of changed material on ability to do formal syllogisticreasoning. Archives of Psychology, 1928, 102, 83. WINTHROV,H. Semantic factors in the measurement of personality integration. Journal of Social Psychology, 1946, 24, 149-175. WOODWORTH, R. S., & SELLS, S. B. An atmosphere effect in formal syllogistic reasoning. Journal of Experimental Psychology, 1935, 18, 451--460.

(Received October 24, 1974)