Propositional logical thinking and comprehension of language connectives: A developmental analysis 9783111352213, 9789027933737

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JANUA LINGUARUM STUDIA MEMORIAE N I C O L A I VAN WIJK D E D I C A T A edenda curai C. H . V A N S C H O O N E V E L D Indiana University

Series Minor, 216

PROPOSITIONAL LOGICAL THINKING AND COMPREHENSION OF LANGUAGE CONNECTIVES A Developmental Analysis

by SCOTT G. PARIS Purdue University

1975

MOUTON THE HAGUE-PARIS

© Copyright 1975 Mouton & Co. B.V., Publishers, The Hague No part of this book may be translated or reproduced in any form, by print, photoprint, microfilm, or any other means, without written permission from the publishers

LIBRARY OF CONGRESS CATALOG CARD NUMBER: 74-75824 ISBN 90 279 3373 1

Printed in The Netherlands by Mouton & Co., The Hague

PREFACE

One of the traditional problems in psychology is understanding how children acquire and utilize abstract relationships. It has been a thorny issue for research in cognitive development as well as for theories of genetic epistemology. Oftentimes, the problem has been approached as the ontogeny of logical thinking. Although this important area of conceptual development has enlightened our understanding of children's comprehension processes, it also has several drawbacks. Most of the tasks used by researchers in analyses of logical thinking have been nonverbal. The subsequently developed models of logical thought are usually extended to include language comprehension, but the validity and utility of this extension remain to be tested. The purpose of the present investigation is twofold: first, to analyze these traditional nonverbal research approaches for their internal validity and inferences to models of logical thinking, and second, to use the empirical research as a direct test of the formal, logical models of cognitive processes with sentential material. The tentative conclusions of this investigation raise serious questions about the explanatory adequacy of present conceptualizations of how children and adults comprehend logical relationships expressed in language. Although the criticisms of traditional research may appear negativistic, they are intended to be constructive and suggestive of more fruitful approaches to the problem. I would like to express my deep appreciation to David Pisoni, Robert Seltzer, Eliot Hearst, and Mel Semmel of Indiana University for their many incisive and thought-provoking comments regarding this research. I am also deeply indebted to Robert Cairns

6

PREFACE

for his counsel and friendship over the years and his critical analyses of issues in developmental psychology. All flaws in this monograph are my own responsibility. This research was supported by a Public Health Service Predoctoral Fellowship PHS 1 FOl MH51274-01 and an Indiana University Grant in Aid of Research to the author. Most of all, I am indebted to Margaret Owen Paris to whom this work is sincerely dedicated.

TABLE OF CONTENTS

Preface

5

List of Tables

8

1. The Experimental Analysis of Propositional Logical Thinking 1.1. Introduction 1.2. General Problem 1.3. The Concept Identification and Attainment Approach 1.4. The Piagetian Approach 1.5. The Psycholinguistic Approach 1.6. Summary

14 34 39 41

2. Empirical Research 2.1. Rationale 2.2. Experiment I 2.3. Experiment II 2.4. Experiment III

43 43 44 61 74

3. General Discussion 3.1. Rejection of the Formal Model 3.2. Towards Alternative Interpretations

82 82 83

Appendices A. Experiment I Test Sentences B. Experiment II Test Sentences C. Experiment III Test Sentences

91 91 93 95

Bibliography Index

11 11 12

97 103

LIST OF TABLES

Table Page 1 Truth Table Representations of Four Common Propositional Rules 13 2 The Logical Truth Table Representations of the Linguistic Connectives 46 3 Percent Errors on Each Connective for Each Grade . . 50 4 Summary Table of the Five Factor Analysis of Variance with Error Scores 51 5 Percent Errors on Each Truth Form Across Connectives for Each Grade 52 6 Percent Errors on Conjunctive Connectives According to Truth Form and Grade 54 7 Percent Errors on Disjunctive Connectives According to Truth Form and Grade 56 8 Percent Errors on the Conditional and Biconditional Connectives According to Truth Form and G r a d e . . . 57 9 Percent Total Responses Conforming to Mismatch Strategy 59 10 Number of Subjects in Each Group Following Particular Error Patterns 60 11 Percent Errors on Each Connective According to Truth Form and Grade 65 12 Summary Table of the Five Factor Analysis of Variance with Error Scores 66 13 Percent Errors by Sex, Grade, and Truth Form . . . . 67

LIST OF TABLES

9

Table Page 14 Percent of Consistent Responses to Two Examples of the Same Propositional Types 68 15 Percent of Consistent Responses to Two Examples of the Same Propositional Type According to Truth Form . . 69 16 Percent Subjects Erring on TF and FT Disjunctive and 69 FT and FF Conditional Propositions 17 Reaction Time Performance of Subjects According to Two Different Criteria 71 18 Average DRT's in Seconds for Each Connective and Each Grade 73 19 Average DRT's in Seconds for Each Truth Form and Grade 73 20 Percent Errors According to Type of Proposition . . . 77 21 Summary Table of the Five Factor Analysis of Variance with Error Scores 78 22 Mean Subjective Evaluation for Each Type of Proposition 79 23 Summary Table of the Five Factor Analysis of Variance with Evaluation Scores 80

1 THE EXPERIMENTAL ANALYSIS OF PROPOSITIONAL LOGICAL THINKING

1.1. INTRODUCTION

During the past fifteen years, psychology has been infused with a vigorous new interest in the mentalistic problems which sparked the birth of scientific psychology in the nineteenth century. It has been suggested that this is a revolutionary shift both in theory and research and not just an ephemeral fashion (Segal and Lachman 1972). Contemporary research is probing areas such as imagery and language processing, which were often considered unapproachable within the constraints of traditional behavior theory. Yet the unobservability of cognitive processes, to which the behaviorists so strenuously objected, still persists. It seems that a primary factor in the renewed interest in mentalistic problems is the general acceptability of broader inferential analysis. Contemporary research regarding cognitive processes characteristically makes inductive inferences to structural models, mathematical models, computersimulation models, and generalized information processing schemata which were not permitted by behavioral accounts. Reliance on inferential methods immediately raises questions regarding the validity of the inferences being made. There are often great distances between performance on laboratory tasks and inferences about cognitive processes. To further our understanding of mental operations it is necessary to critically evaluate the research methodology upon which the inferences are based: we must ask if particular experimental tasks are valid instruments and actually evaluate that which they purport to measure. This is recognized as

12

THE EXPERIMENTAL ANALYSIS

the familiar problem of the correspondence between the experimenter's nominal account and subjects' functional behavior. Nowhere is this problem more visible than in assessments of logical thinking abilities. The problem that confronts us now is to analyze the research paradigms and methods in order to determine whether or not they support the inferences made to rule-following cognitive processes. We shall look at three separate approaches to the problem : conceptidentification research, Piagetian theory, and comprehension of language connectives. It should be explicitly understood that the aim is NOT to evaluate applications of logic to language or experimental tasks but rather to investigate how people understand the rule relationships of propositional logic. We are interested in cognitive, rule-following, problem-solving processes.

1.2. G E N E R A L P R O B L E M

By far the most well-known and researched system of mathematical logic is propositional logic. The presence and absence of two events A and B can be combined in four ways: AB, AB, AB, and AB. The presence of an event is denoted as True (T) and the absence as False (F). To those familiar with logic, these four permutations are recognized as the index of the logical truth table and hereafter will be referred to as truth forms: TT, TF, FT, and FF. The logical relationships of the four truth forms can be combined in sixteen unique bidimensional partitions, thus forming the calculus of propositions (cf. Bourne 1970: 554). Each of these partitions can be characterized by a logical rule. For example, when the TT form is true and all other forms are false, we say the events described by T T are related by the rule of conjunction. Some of the more common logical rules are conjunction, disjunction, conditionality, and biconditionality. These relationships are shown in the truth table in Table 1. The calculus of propositions is the formal model of the tasks used in research regarding propositional logical thinking. Bourne (1970)

T H E EXPERIMENTAL A N A L Y S I S

TABLE 1.

13

Truth Table Representations of Four Common Prepositional Rules

Propositional Rule

Attributes (Truth Forms)

Verbal Relationship

AB AB AB AB (TT) (TF) (FT) (FF) Conjunction Disjunction (Inclusive) Conditionality Biconditionality

T T T T

F T F F

F T T F

F F T T

A and B AorB if A, then B if and only if A, then B

offered a conceptual framework which clarifies the three possible types of problems researchers use. By designating the events A and B (and their absence) as ai and a2, and a logical rule as R, Bourne distinguished between rule learning, where subjects must discover how the attributes are related (ai ? and attribute identification, where subjects must identify the correct attributes which satisfy the given rule (? R ?). The difference between the two tasks is simply which information is given to subjects — the relevant attributes or the rule. In contrast to attribute-identification and rule-learning tasks, during complete learning the experimenter furnishes the subject no initial information and the task requires discovery of the relevant attributes and the rule. There exists another potential task paradigm, however — presenting subjects with ai R and requiring a judgment of "True" or "False" of the given relationship. This will be referred to as an evaluation task. A variety of tasks and problems conceptually conform to the logical model and these four experimental paradigms. The ability to solve these problems has been used as a measure of logical thinking abilities and higher order cognitive processes. The subsequent section reviews and evaluates the major research contributions with special attention to the validity of measures, methods, and inferences.

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THE EXPERIMENTAL ANALYSIS 1.3. THE CONCEPT IDENTIFICATION AND ATTAINMENT APPROACH

This approach is by far the most popular and most prolific contemporary method for investigating propositional thinking. The implicit assumption is that the rules of propositional logic are concepts and may be studied in a fashion similar to other conceptidentification and attainment problems. For example, in an attribute identification task, one might be investigating the identification of the concept's redness and triangularity. The research of Bourne and his co-workers is an excellent example of a typical experimental paradigm and should elucidate the usual methods and outcomes characteristic of this approach (Bourne 1970). 1.3.1. A Typical Problem Research subjects are usually college students, although children have sometimes been tested. The task is almost invariably a cardsorting problem. Each card in a deck or an array has a variety of dimensional features such as color, shape, size, and background. There can be any number of stimulus dimensions and any number of attributes per dimension. Frequently, four dimensions, with three attributes on each, are used to develop the stimuli which results in eighty-one different cards. In a rule learning task the subject must sort these cards into two categories labeled YES and No or POSITIVE and NEGATIVE to indicate whether or not the example conforms to the rule. The instructions define the task, usually as rule learning or attribute identification, and appropriate practice with rules or instances follows. After this pretraining, testing is carried on until a criterion is reached. The criterion is often an errorless run of ten or sixteen trials, but is frequently the correct verbalization of the rule or relevant attributes by the subject. Although sixteen rule combinations are possible, research usually is confined to conjunction, disjunction, conditionality, and biconditionality (Bourne 1970; Haygood and Bourne 1965). The usually obtained order of difficulty for

THE EXPERIMENTAL ANALYSIS

15

card-sorting tasks is exactly as stated above, with conjunction being the easiest logical relationship to master. The task often involves repeated testing with each subject solving several problems. This can be a test of either inter- or intrarule transfer. Following several problems, the initial rule differences disappear and each successive problem is solved rapidly. One interpretation suggests that subjects acquire an intuitive truth table strategy to solve the problems and that the initial rule differences are a function of the compatibility of each rule to this strategy. The truth table strategy has two parts: first, the differentiation of the stimulus array into the four truth table classes — TT, TF, FT, and FF — which signify the presence and absence of the relevant attributes; and second, the assignment of each of the truth forms to one of the two response categories. Bourne incorporated these steps into a hierarchical model of cognitive, logical processing of information (see Bourne 1970: 555). These findings are highly reliable and have been replicated many times in the literature. Recently, the results have been demonstrated in a developmental study, and it has been suggested that children as young as six years of age solve these problems in an analogous truth-functional manner. 1 The facilitative effects of previous rulefollowing experience have been amply demonstrated by many transfer studies (Bourne and Guy 1968a; 1968b; Bourne 1970). In fact, it was found that pretraining with the logical truth table facilitated performance whereas merely describing the rule did not (Guy 1969). In summary, the card-sorting procedure for assessing propositional rule learning has been extensively and indeed almost exclusively employed. The research results have shown great consistency and the interpretations have not been challenged. The following two sections offer theoretical and methodological evaluations of the concept identification approach. 1

"On the surface, the foregoing interpretation of rule difficulty suggests that children, just as adults, respond to stimuli as members of the coded classes prescribed by the truth table. That possibility, in turn, implies that children as young as 6 years have the coding skills required for efficient problem solving within the calculus of propositions" (Bourne and O'Banion 1971:532).

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THE EXPERIMENTAL ANALYSIS

1.3.2. Theoretical Critique The critical question to be asked of the concept identification paradigm concerns the justification of inferences regarding logical, rule-following behavior from card-sorting behavior: "Can the rule learning task be solved by subjects independently of the logical rules which characterize the task?" 2 Two quotations regarding the truthtable strategy are particularly useful in attempting to answer this question. Once the strategy is mastered it results in a marked simplification of RL problems even if the rule is new or unfamiliar, for the task merely requires S to learn the assignment of these four classes to two response categories (Bourne and Guy 1968a: 424). Given this principle, any problem based on a new or an unfamiliar rule becomes an almost trivial paired-associates task, requiring only a few trials to master (Bourne and Guy 1968a :429). This simplistic paired-associate type of learning appears to be a possible and likely description of subjects' problem-solving behavior during card-sorting tasks. Any problem of the sort administered in this experiment has a concrete solution in the sense that a particular, delimited population of stimulus objects is simply divided into two groups. The subjects can achieve that solution without learning an abstract rule, generalizable over many class concepts. On the other hand, if the subject does learn a rule which is more general than one specific solution, for example, if he learns how to conjoin two attributes X and Y with unspecified values, that knowledge should facilitate performance in subsequent similar problems (Bourne and O'Banion 1971: 527). A perplexing contradiction is now evident. On the one hand, Bourne asserts that the card-sorting tasks may be solved independently of knowledge of the abstract propositional rule, while, on the other hand, conceptual rules of propositional calculus are inferred to order subjects' behavior. This is especially true for inter2 The differentiation of the nominal model and the functional behavior is critical in the ensuing discussion.

THE EXPERIMENTAL ANALYSIS

17

problem transfer where the rule is applied to different stimuli. But why can't the paired-associate rule generalize instead of the abstract rule to control performance? The critical question is, "Are paired-associate learning and propositional rule following the same cognitive process?" At this point, the truth-table strategy outlined by Bourne does not appear to justify propositional rules as the cognitive processes involved in task solution; consideration of the subject's behavior during a rule-learning problem will illustrate this. Confronted with a deck of cards or an array, the subject must discover how two relevant attributes are related, let's say redness and circularity. Suppose that the subject has progressed through the first step of the problem-solving process suggested by Bourne and has categorized the stimuli as red, circle; red, noncircle; nonred, circle; nonred, noncircle; i.e. the four logical truth forms. Now suppose that over trials he learns to place all cards except red, noncircle cards (TF) in the positive category. At this point, is it judicious to assert that he knows the rule, "if the event is red, then it is circular"? The methodology asserts that the subject has "learned" the logical relationship of the rule of conditionality and has the CONCEPT of the rule. However, the subject could solve this task in a paired-associate manner by placing all red noncircular cards in one category and all others in the other category with no consideration of the logical rule. The paired associates here are not specific cards and response categories but, as Bourne suggested, permutations of the relevant attributes, i.e. truth forms and response categories. This conceptualization of the paired items actually frees the explanation from a strict stimulus-response interpretation and requires categorical abstraction of the stimulus features by the subject. However, this is not too difficult. Consider the case where the subject is informed that circularity and redness are the relevant attributes. Red triangles, red squares, red stars all belong in the same category. Similarly, blue stars, blue squares, and blue triangles can all be grouped as FF instances. The reduction of the stimulus array into four classes which conforms to the four truth forms is certainly plausible and an interesting abstracting ability. However,

18

THE EXPERIMENTAL ANALYSIS

the systematic pairing of one of these classes with a response category simply does not involve propositional rule-following behavior on the part of the subject. The problem centers on the notion of a rule. On the one hand, we have a formal propositional rule of logic which is the structural description of the task, and on the other, we have a response assignment rule which describes the subjects' card-sorting behavior. When the subject has met the criterion and solved the task, what does he know? Bourne suggested that subjects learned the response assignment rule (i.e. the paired-associate strategy) AND the corresponding implicit formal rule. Although this is possible, it has not been demonstrated within this card-sorting paradigm. It has only been shown that subjects learned a response-assignment rule which is formally isomorphic with a rule of propositional logic. The choice of a formally similar rule is totally arbitrary. Although one might say that sorting all cards into a positive category except TF cards demonstrates the knowledge of the relationship A IMPLIES B, it could as easily indicate the relationship A OR B, which is a logically equivalent expression. There is no reason to assume that the subject learns ANY formal rule as a necessary consequence of learning the response-assignment rule, much less a particular rule which was picked as an a priori structural description of the task. The solution of the task and knowledge of the response-assignment rule in no way implies the subjects' knowledge of abstract rule relationships. Interpretations of card-sorting behavior in terms of "intuitive truth-table strategies" are unwarranted and gratuitous. As long as the subjects' task-solution behavior can be parsimoniously and adequately explained by a type of paired-associate learning, reference to more abstract explanatory frameworks is not required. A further difficulty with the sweeping inferences is their reliance on a single methodology. The card-sorting tasks which confuse formal and response-assignment rules have been virtually the only tasks used in the experimental analysis of propositional rule relationships.3 If one hypothesizes that knowledge of the formal rule is 3 Again, it must be reiterated that our goal is to understand the cognitive processes underlying comprehension of propositional rule relationships. It is

THE EXPERIMENTAL ANALYSIS

19

implied by knowledge of the response-assignment rule, as Bourne did, it appears incumbent to demonstrate that knowledge with other tasks in different situations. Although the paired-associate interpretation appears to be an accurate description of subjects' response-assignment rules, it is not without complications. One problem is that it predicts equal learning facility among different rules and logical relationships. Equal facility, though, occurs only after several problems and presumable refinement of the truth-table strategy. Bourne and O'Banion (1971: 531) suggested two explanations for differences in initial rule-learning difficulty. One possibility is a hierarchical organization of logical relationships similar to that postulated by Neisser and Weene (1962). The other, which they prefer, is that subjects come to the laboratory with preexperimental experience and an inclination toward solving for conjunctive concepts (cf. Bruner, Goodnow, and Austin 1956). The suggestion was that subjects have "habits" which dictate sorting (a) TT stimuli into the positive category, (b) FF stimuli into the negative category, and (c) TF and FT stimuli into the same category. The mixed truth forms would be assimilated into the positive or negative class depending on "the relative strengths of the generalization gradients" (Bourne and O'Banion 1971: 532) which are a function of the number of TT and FF stimuli in the sample. In this manner, Bourne and O'Banion accounted for the initial learning differences among rules. However, the origins of these habits are left unspecified and the associate learning — four truth forms to two response categories — remains.4 Although the explanation fits the data, the inferences in terms of logical ruleacknowledged that formal logic often provides nominal or definitive task structure and the use of formal rules is quite legitimate for this purpose. However, the extrapolation of formal rules as actual cognitive and response assignment rules is unwarranted within the constraints of card-sorting tasks. 4 Of course, the ontogeny of these "habits" is extremely interesting and was examined in depth by Bruner, Goodnow, and Austin (1956). It would appear more fruitful to inquire as to the nature and development of these "habits" rather than concentrate on the outcomes in terms of formal rule differences which are only artifacts. Rule differences are obtained because of the interaction between predispositions and the paired-associate strategy and not because of the way people understand different logical relationships.

20

THE EXPERIMENTAL ANALYSIS

following behavior are again gratuitous. The entire task appears accounted for by paired-associate learning and biasing habits. Only the rule-learning task has been discussed so far. The attribute-identification task always involves an explanation of the rule to be employed and usually is accompanied by practice or pretraining. Because of this, it has little to do with how people normally learn or comprehend these rule relationships. However, it does offer valuable information regarding subjects' ability to use these rules in categorization tasks. But we can ask if these tasks demonstrate knowledge of logical relationships and logical rule utilization or some other type of problem-solving strategy. Again if we look at the subjects' task, the difficulty becomes more apparent. Imagine that a subject was pretrained on a conjunctive concept or rule. He is told that the test will involve the same rule but his job is to find the relevant attributes. He is next shown a card from the stimulus array and is usually told that this card is an example of the concept. The subject's task is then to determine what aspects of the exemplar are important and related by the conjunctive rule. Several important variables are involved in this procedure, though, other than the subject's ability to use conjunction. These variables, including the number of dimensions and attributes in the task, the salience of particular attributes, the individual's ability to form hypotheses, and his memory for the outcomes of the tests, are extremely important for other rules in attribute-identification tasks, such as disjunction, which often is initiated by an example card of what the concept is NOT. These comments anticipate the methodological critique section, but it is important to point out that both rule-learning and attributeidentification tasks may be highly constrained by the card-sorting methods. Put more strongly, subjects involved in typical card-sorting tasks may respond and solve the problems in a paired-associate manner INDEPENDENTLY of the logical rules which nominally describe the task. The distinction between the nominal model of the task and the functional behavior by which the problem is solved is the heart of the issue. As long as rule-learning tasks can be potentially, and Bourne suggests ACTUALLY, solved by paired-associate

THE EXPERIMENTAL ANALYSIS

21

learning which does not necessarily imply knowledge of the formal rule, then there should be caution in accepting inferences to logical rules and formal models of subjects' problem-solving behavior. Similarly, the solution of attribute-identification tasks may also rely on types of behavior for problem solution other than the nominal logical processes. One of the contentions of this paper is that the highly constrained methods of assessing propositional logical thinking in concept identification research have led to injudicious inferences and acceptance of a logical model which nominally describes the task but obfuscates understanding of the processes by which subjects solve the problems. 1.3.3. Methodological Critique Truth forms. Van Rijn (1971: 16) has suggested that the learning differences among rules may be more appropriately considered as learning differences among truth forms. Some truth forms may simply be more difficult to learn or categorize and various rules differ in regard to which truth forms are positive and negative. This seems a plausible interpretation and similar to the differences among truth forms due to conjunctive sets and biasing habits. Under these circumstances, it is perplexing why card-sorting tasks do not contain equal number of instances representing the various truth forms. In the typical task, the proportion of truth forms, TT: TF: FT: FF, is 1:2:2:4 (Bourne 1970: 549). It appears that this is a critical feature of the task and can contribute to initial learning differences among rules. For example, let's consider the biconditional and conditional rules. Both of these rules require TT and FF instances to be placed in the same positive category in order to be correct. This is contrary to the subjects' preexperimental habits of sorting TT's into the positive category and FF's into the negative category. Not only is the subject working against his biases on sorting the FF instances, he also encounters four times as many of them as the TT instances. Therefore, one would expect biconditional and conditional rules to be the most difficult (as they are) for two reasons. First, subjects should have more trouble sorting each FF card because it contra-

22

THE EXPERIMENTAL ANALYSIS

diets his biases. Second, it will be more difficult for subjects to achieve criterion performance because the stimulus deck contains more troublesome FF cards than easier forms. In order to solve conditional and biconditional rules, subjects must unlearn their biases. Presumably, that is why pretraining on conditional problems has such a beneficial transfer effect on subsequent biconditional rulelearning problems (see Bourne and Guy 1968a). Similar problems exist for other rules. For example, the mixed truth forms in a disjunctive problem are usually the most difficult to master and they each occur twice as often as the relatively easy TT form. The interactions among rules, truth forms, and biasing habits are subtle yet powerful in their effects. The danger of making inferences and claims about logical problem-solving processes is again evident if the data can be accounted for by preexperimental habits, unequal truth-form representations, the interaction of these two factors, and the paired-associate learning. Although the latter explanation may be more difficult to describe, it appears much more accurate and parsimonious. It is unnecessary to appeal to abstract structures and explanations when the behavior can be adequately accounted for by lower level descriptions. The solution to the problem of equal truth-form representation is not easy. With several dimensions and several attributes per dimension, there are an unequal number of possible permutations of truth forms. If one equates numbers of truth forms, one must drop certain stimulus combinations from the array. This has potential shortcomings because of potential differences in salience and difficulty between stimulus cards. Equalizing the number of truth forms also causes an imbalance in the number of positive and negative instances for most rules. Likewise, one cannot equate positive and negative instances across rules because this leads to possible differences between numbers of truth forms depending on the rule. In fact, for most rules one must trade off equal numbers of truth forms or equal numbers of positive and negative instances. The effects of each profoundly influences card sorting. No solution is offered to this methodological problem but it is hoped that recognition of the difficulty will clarify the constraints of card-sorting tasks and illus-

THE EXPERIMENTAL ANALYSIS

23

trate how initial rule-learning differences may be obtained in a traditional card-sorting task. Instructions. Mention has already been made of possible differences between positive and negative instances in their effects on rulelearning and attribute-identification tasks. These issues will be extensively discussed later, but at this point it is sufficient to note that subjects gain different amounts of information from positive and negative instances depending upon the particular rule. Further, most subjects show a preference for positive instances and problemsolving strategies based on positive instances (cf. Bruner, et al. 1956; Wason 1959). One problem with these differential capacities is the subtle influence of the task instructions and procedure. Specifically, there are vast differences in performance with a rule depending on whether the subject is shown a positive instance exemplar or negative instance nonexemplar for a rule-learning task. For example, if the subject is trying to learn the conjunctive rule red and square, he is typically shown such an instance and told, "This card follows the rule." For a similar disjunctive task, red or square, the subject is usually shown a negative instance (maybe a yellow circle) and told, "This card does not follow the rule." Even though the two tasks are equally structured and nominally equivalent in information, reasoning from positive instances is best and the tasks differ in difficulty (Conant and Trabasso 1964). It appears that there is a biasing factor towards focusing on positive instances which may be a preexperimental and general bias, but the instructions also foster a positive focusing strategy because they direct subjects to find the instances which FOLLOW a particular rule or to discover the rule. This bias was illustrated in a study by Snow and Rabinovitch (1969). Their subjects solved conjunctive or disjunctive rule learning tasks (although there was considerable rule training) and could select the cards from the total population. The researchers tabulated whether the first choice by subjects (after an exemplar for a conjunctive task and a nonexemplar for a disjunctive task) was a positive or negative instance. Summing over five age levels, there were 180 positive instances chosen on the conjunctive

24

THE EXPERIMENTAL ANALYSIS

task and fourteen negative instances. However, for the disjunctive task, there were 111 positive instances chosen and eighty-three negative instances. Despite the fact that a negative disjunctive instance was the most ideal choice, subjects usually selected positive instances. It seems plausible that if subjects are directed to find the rule they will indeed concentrate on positive instances. For the subjects in many studies, especially those comparing conjunctive and disjunctive rules, this biasing factor is debilitating to their performance. Clearly, the obtained differences (e.g. Snow and Rabinovitch 1969) are not due solely to differences between logical rules and subjects' cognitive processes but arise as procedural artifacts. This bias is also perpetuated by the categories into which the stimuli are sorted. They are typically labeled G O O D and BAD, or POSITIVE and NEGATIVE — not simply A and B . The only study directly concerned with this problem found that subjects performed better under concept learning instructions which de-emphasized the positive-negative dichotomy of the categories (Denny 1969). Another potential problem with the procedures of concept identification tasks is the training. Oftentimes the instances, dimensions and rules are explained and pretraining problems are given, especially in attribute-identification tasks. It is immediately obvious that the tasks are not assessments of normally applied processes. A potentially more serious problem, however, is the damage to the credibility of inferences made from the task. It is quite possible that differential rule-learning or attribute-identification abilities merely reflect differential training. Do two conjunctive warm-up problems raise performance to the same level as two disjunctive warm-up problems? Are the relationships of a conjunctive rule more easily verbalized, trained, and understood than other rules? These kinds of questions again raise the possibility that obtained differences with logical rules are the result of specific task procedures and not subjects' abilities to process logical relationships. Stimulus dimensions and attributes. The decisions regarding the

actual stimuli to be used in the task are generally governed by tradition. The dimensions most often used are color, shape, and number, although others are possible. The assumption of equal

THE EXPERIMENTAL ANALYSIS

25

attention-eliciting properties of the various dimensions and attributes is common. This seems strange in those studies where cues involve subtle background changes in color or texture, or complex figure and number arrangements. Traditional use should not lead researchers to ignore the potential problem of inequity among stimuli. A more serious problem concerns the number of dimensions and attributes which are employed. The minimum number required for the four truth forms is two dimensions with two attributes on each. Most studies usually add one or two irrelevant dimensions and include at least three attributes on each. The problem of increasing numbers of dimensions and attributes is that the permutations do not yield equivalent numbers of truth forms. This factor was responsible for the 1:2:2:4 unequal truth-form representation. Laughlin (1968) points out that one consequence of this is that only two-valued dimensions allow subjects to draw inferences from an attribute's absence. Increasing the number of attributes actually changes the amount of information which can be effectively drawn from the stimulus population. Of course, increasing the number of dimensions and attributes greatly increases the number of difficult FF truth forms in the population, too. Johnson, Warner, and Lee (1970) required subjects to point at the relevant values of each stimulus card and found that enforced attention improved performance on FF forms, presumably because of decreased attention to irrelevant stimuli. It should also be noted that the FF forms are the most variable truth form. Not only are there more FF forms in the population than any other, but they are comprised of more variable stimulus combinations. It is no wonder that concentration on the relevant attributes (and their absence) results in better performance on these forms. Coupled with the extreme difficulty of FF forms on conditional and biconditional rules, this evidence again suggests that some of these logical relationships are more difficult to learn because of procedural artifacts which increase the occurrence of their most difficult components. Schwartz (1966) found that 75 percent of subjects solving dis-

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THE EXPERIMENTAL ANALYSIS

junctive complete learning tasks had to eliminate all erroneous hypotheses before a solution could be achieved. In the disjunctive relationship, this means that subjects had to learn all the negative instances, which were all the FF forms. It is obvious that this would be more difficult with increasing numbers of dimensions and attributes because more negative instances and erroneous hypotheses would be available. Not only would the problems constitute more hypothesis testing, but they would also add increased memory load if the results were not displayed or available. The net result of increasing the number of dimensions and attributes in a card-sorting task is an unequal increase in numbers of truth forms, with the more difficult forms being overrepresented. This consequently causes poorer attention to the relevant stimuli, more hypotheses to be tested, and increased memory demands. These constraints on subjects' cognitive processes arise not merely as a function of logical rule difference but as a consequence of the particular card-sorting methods employed. Verbalization as a criterion. Another problem with concept identification research methods is determining when subjects have learned the concept. A frequent measure is the correct sorting of many cards; oftentimes as many as sixteen successive correct card sorts. A different criterion employed by many researchers is the verbalization of the correct rule or attributes in the task (see Conant and Trabasso 1964; Gardner 1970; Krebs and Lovelace 1970; Laughlin 1968; Schwartz 1966). Verbalization of the correct rule in rule learning tasks seems to be a weak criterion, though, especially when comparing the relative difficulty of rules. First, there is the possibility that subjects can effectively use the rules but simply cannot verbalize them. For example, Wells (1963) found that training on disjunctive problems facilitated the choice of a disjunctive rule to describe an array which could be defined either conjunctively or disjunctively. But this facilitation was in contrast with a control group who never offered a disjunctive rule choice. One can ask if the pretraining actually changed subjects' concepts of the array or only altered the probability of eliciting a verbalization of the disjunctive rule. It may be that dis-

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junctive relationships are difficult or uncommon for subjects to verbalize and therefore are shunned. There is little reason to expect differential preference or availability of verbal rule descriptions to correspond to different cognitive abilities to utilize or understand the relationships. A second problem concerns the verbalization of incorrect hypotheses or wrong guesses of the rule. Some researchers have used no feedback at all (Krebs and Lovelace 1970) and others have used a correction procedure (Schwartz 1966). Conant and Trabasso (1964) told subjects that incorrect hypotheses would result in a subtraction from their score. There is considerable variability in the use of this criterion among investigators which directly causes variability in the number of hypotheses offered and the probability of subjects offering one. If the subject has to verbalize a rule after every trial and obtains feedback, it seems that he has a high probability of merely guessing the correct rule. This is especially true for those studies using few stimulus attributes and dimensions and where the population of rule hypotheses entertained by the subject is small. The number of incorrect guesses and their content is virtually never reported in the research report. It would be interesting to know if subjects adopt a strategy of guessing attribute combinations like red-square, yellow-square, blue-square, red-circle, etc., after every trial. Besides the number and content of the guesses, it would be interesting to know the decision rules under which subjects guess. For example, disjunctive concepts may be less common (Bruner, et al. 1956), and subjects may test out unfamiliar hypotheses more carefully and redundantly than a more familiar conjunctive rule. At any rate, it is certainly possible that subjects will not offer different hypotheses equally often under various payoff conditions, especially when those hypotheses vary in familiarity, availability, or difficulty. The potential problems with verbalization as a criterion again suggest that artificial differences among logical relationships exist as a function of methodological factors rather than differences within subjects' cognitive processes.

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1.3.4. A Positive Perspective The concept-identification research has been criticized for its invalid inferences to complex, logical, cognitive abilities of subjects in these highly constrained tasks. 5 Although these inferences seem unwarranted, that is not to say the research provides no useful information. On the contrary, understanding of cognitive processes has progressed notably in two distinct areas as a consequence of concept-identification and logical rule research. Those areas concern (a) the use of problem-solving strategies and (b) the role of positive and negative instances during problem solving. The major research results in these areas will be briefly discussed. Problem-solving strategies. Undoubtedly the most comprehensive analysis of strategies used to solve attribute-identification and rulelearning tasks for conjunctive, disjunctive, and probabilistic relations was expounded by Bruner, Goodnow, and Austin (1956). Periodic reference to that work has been made in the foregoing pages. Their analysis was concerned with how information is obtained and used with particular attention to memory load, features of the stimulus instances, and possibilities to draw inferences from the available information. The major strength of this research is the empirical identification of strategies used by subjects to solve card-sorting tasks. (The authors, interestingly enough, called this category learning and were not influenced by a priori logical models.) They postulated four discernible strategies by which subjects could solve conjunctive attribute-identification tasks and a brief elaboration is illustrative of their approach. SIMULTANEOUS SCANNING refers to a strategy whereby each encountered instance is used to deduce which hypotheses are tenable 5

The methodology was critically evaluated; however, any research design in experimental psychology has potential shortcomings. The critique is not intended to be a devastating blow to the paradigm nor a tedious harassment. The purpose is to explicate how rule differences may be obtained within the paradigm due to methodological and procedural variables. It is felt that a careful analysis reveals little substantive support for the broad inferences to cognitive prepositional rule following or intuitive truth table strategies. It is the inferential analysis made from the task rather than the task itself which is criticized.

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and which are eliminated. Each successive instance is selected to test new hypotheses and avoid redundancy. This strategy is maximally informative but exacting. It requires subjects to retain a tremendous amount of information in memory and proceed through many complex inferences simultaneously. A SUCCESSIVE SCANNING STRATEGY requires subjects to test a single hypothesis at a time. The subject formulates an hypothesis and tests it DIRECTLY with each new card selection. As a consequence, this strategy involves testing redundant stimuli and is not as efficient as simultaneous scanning. The advantage of successive scanning is almost entirely due to less strain on memory and the reduced number of inferential operations employed at the same time. Bruner, et al., emphasize that this gain is primarily due to the fact that subjects test hypotheses DIRECTLY, that is, with a positive instance of that stimulus configuration. The strategy does not provide maximum information per choice nor does it regulate risk taking, but it is useful "when the cognitive going gets rough or when one has good reason to believe that a particular hypothesis will turn out to be correct" (Bruner, Goodnow, and Austin 1956: 87). CONSERVATIVE FOCUSING is a strategy which focuses on a positive instance and proceeds to make sequential choices which alter one attribute of the focus card and determines if each test results in a positive or negative instance. The successive choices are never redundant and always provide some information to subjects. Again, cognitive economy is maintained by ordering the stimulus population, systematically eliminating attributes, and directly testing each new hypothesis. Focus GAMBLING is a strategy in which subjects focus on a positive instance and change more than one attribute in each card selection. Although this strategy is economical like conservative focusing, it involves greater redundancy and risk factors. For example, subjects obtain little or no information from negative instances. The authors discuss many variables which influence the employment of these strategies such as type of instances, order of the array, and memory requirements. The analysis always focuses on the processes by which subjects perform on these tasks. Some of their

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conclusions will illustrate the generality of their process-oriented explanations. Bruner, et al., noted that subjects are reluctant to use disjunctions and often attempt conjunctive solutions, regardless of the problem. There is a corresponding preference for direct testing from positive instances as opposed to indirect testings with negative instances. This is particularly debilitating for disjunctive problems where indirect inferences from negative instances must be drawn. The difficulty is subsumed under the general strategy of POSITIVE FOCUSING and the preference for making inferences from the whole instance as opposed to its parts. Historically, the work of Bruner, et al., is the most significant analysis, although other explanations have been advanced. Haygood and Bourne (1965) reviewed four hypotheses to account for differences among logical rules in card-sorting tasks. The first emphasized a predilection for conjunctive groupings on the basis of experience. The second hypothesis regarding differential familiarity was highly similar. Both of these had already been suggested by Bruner, et. al., in 1956. The third hypothesis, proposed by Neisser and Weene (1962), suggested that there were three levels of rule difficulty based on the number of operations required by each rule. Level I was affirmation or negation of a single attribute; Level II was the group of conjunctive, disjunctive, and conditional operations. This hypothesis has not been favorably received and does not account for the obtained results with rule difficulty. The fourth hypothesis, suggested by Haygood and Bourne (1965), was based on the differences in information and uncertainty afforded by the splits of truth forms into positive and negative classes for various rules. This type of hypothesis was the predecessor for the paired-associate learning model and the later addition of "biasing habits". An even more recent hypothesis to emerge from Bourne's laboratory is the suggestion that subjects combine attributes into natural adjectivenoun sequences and then focus on the noun's attribute assignment (Van Rijn 1971). This is the first linguistic hypothesis to emerge from these nonverbal tasks and represents a departure from the other proposed strategies.

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In sum, these various hypotheses represent valuable information for research and understanding of the behaviors by which subjects solve these tasks. The brief discussion of problem-solving strategies emphasizes some of the positive gains to be found in concept-identification research. However, it will be noted that these strategies do not necessarily conform to the propositional logical operations which characterize the task. The fact that subjects can solve these tasks without knowledge of the rule and its logical relationships is devastating to the inferences made from the tasks to "intuitive truth tables" but not to the usefulness of the tasks themselves. The role of positive and negative instances. I n card-sorting tasks,

positive and negative instances play an important role. This has been mentioned throughout and the following comments will hopefully tie together the main points regarding this factor. Bruner, et al., emphasized the predisposition of subjects to sort according to conjunctive strategies. Similarly, Bourne and O'Banion's (1971) emphasis on "biasing habits" reflects subjects' predilection to approach the task with a conjunctive set. The unspecified bias towards conjunctive solutions and preference for the positive information in TT forms is one of the most consistent findings in the literature. A natural correlation of this bias is a positive focusing strategy. King (1968) suggested that subjects go through two steps: first, subjects focus attention on the positive class, and second, they assume the presence of relevant attributes indicates a positive instance. This type of process is almost matching to sample for the conjunctive TT form. There are numerous accounts of the greater difficulty in focusing on negative instances and solving disjunctive problems (see Conant and Trabasso 1964; Donaldson 1959; Gardner 1970; King 1966; Wason 1959). This would not ordinarily be expected because the TT conjunctive positive instance is informationally equivalent with the FF disjunctive negative instances. Schwartz (1966) demonstrates how the positive focusing strategy can be counter-productive: he found that subjects who selected their own cards in a disjunctive task had choices WORSE than chance due to the consistent selection of positive instance cards.

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There appears to be great reluctance of subjects to attend to negative instances and make the required indirect tests of hypotheses. Bruner, et al., noted the preference for direct testing and avoidance of the extra transformation involved with indirect procedures. Bourne, Ekstrand, and Montgomery (1969) found that although the truth forms were represented in the 1:2:2:4 ratio, subjects chose 40 percent TT, 22 percent TF, 21 percent FT, and 17 percent F F instances on an attribute-identification task. This shows a marked preference for direct testing with positive instances and attributes' presence. However, Seggie (1970) cautioned that reluctance does not mean a processing inability; he found that some subjects used indirect tests of hypotheses. The best reason for avoiding negative instances and indirect hypotheses seems to be cognitive economy or less strain. Several studies have shown that when subjects can inspect earlier instances and outcomes and do not have to rely on memory, they select more negative instances and do better with them (Bourne, et al., 1969; Cahill and Hovland 1960). Braley (1963) suggested that subjects confronting negative instances can either discard them or store the instance and its negativity. He asserted that this is a tremendous memory load and an unusual function for negative instances which usually disconfirm hypotheses and do not take part in the searching and testing of cue information. Perhaps the effects are best summarized by two quotations (Nahinsky and Slaymaker 1970: 67,68). The results suggest that Ss shift to a focusing strategy upon the first positive instance and abandon the attempt to use information from prior negative instances at that point. The information provided by a negative instance alone is meager in comparison with the burden placed on memory in processing it. It appears that pure logic and information content of stimuli must be combined with other factors to determine the solution process. Strategy choices which conserve storage capacities are necessary for efficiency.

Summary. In conclusion, valuable information has been obtained from card-sorting tasks regarding the utility of positive and negative instances and inferences which subjects draw from them as well

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33

as hypothesis-testing strategies of subjects. The complexity of these cognitive operations, though, clearly indicates that formalistic problem-solving analogues do not adequately describe subjects' behavior when solving these tasks. It seems that the methodological constraints of card-sorting tasks have given a narrow view of subjects' logical rule learning. An adequate conception of the issue must go beyond formal notions like the intuitive truth-table strategy and consider the many other strategies and processes (e.g. focusing, memory load, direct hypothesis testing) which appear vital for task solution. In this regard, two areas were briefly discussed — problem-solving strategies and the role of positive and negative instances — as illustrative of the positive gains made from cardsorting tasks. In a similar manner, there has been valuable information obtained in Bourne's research concerning the paired-associate strategy and abstraction of stimuli into conceptual classes similar to truth forms. This research and the developmental implications are important and useful without recourse to formal analogues. A basic theoretical problem in this research concerns the postulation of a cognitive process labeled propositional logical thinking. Although such an ability has been tacitly assumed in the past, there are at least three disadvantages with it. First, we have rather arbitrarily divided cognitive abilities into separate entities on the basis of a priori mathematical systems. Should one also consider separate cognitive processes such as predicative logical thinking, Euclidean geometric thinking, non-Euclidean geometric thinking, etc.? This partitioning does great injustice to the individual and to our knowledge about general cognitive principles. The second point concerns the attribution of a particular cognitive skill to a subject who solves a task which conforms to the formal structure of that skill. That solution of a propositional logical task implies propositional logical thinking is obviously circular. Further, one can ask whether we should attribute propositional logical thinking to nonhuman species who solve formally identical problems. The differentiation of response assignment and formal rules is crucial here. It must be emphasized that knowledge of response assignment rules which are essential for task solution may or may not also imply knowledge of

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the formal rules. Current research reviewed here has demonstrated that paired-associate learning adequately characterizes subjects' response-assignment rules during card-sorting tasks. However, there is no substantiation for concomitant, implicit knowledge of the abstract rule relationship which may nominally describe the structure of the task. The third problem is closely related to the other two and involves the problem of reification (see Eacker 1972). Hull (1943:28) defined the issue by saying, to reify a function is to give it a name and presently to consider that the name represents a thing, and finally to believe that the thing so named somehow explains the performance of the function Unfortunately, this sequence describes the research involving propositional logical thinking. Formal propositional logical thinking has indeed been advanced as a functional explanation of cognitive processes with which it originally shared only a formal structural similaiity. The intuitive truth-table strategy is nothing more than a reified formal analogue of propositional logic which is offered as a problem-solving, behavioral explanation. This obfuscates our understanding of the cognitive processes involved. Perhaps the discrimination between response-assignment rules and formal rules, the renunciation of a reified formal logic as a cognitive explanation, and illumination of the methodological problems associated with card-sorting tasks will help dispel this compartmentalized view of logical thinking.

1.4. THE PIAGETIAN APPROACH

One can hardly discuss logical thinking without discussing the work of Piaget who was perhaps the first contemporary psychologist to concern himself with the cognitive processes underlying propositional reasoning. A synopsis of Piaget's view of propositional logical thought must encompass previous development. The child passes through the sensorimotor and concrete operations stages during which he assimilates vital information. Through the joint

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processes of accommodation and assimilation the child's cognitive structures undergo continuous change. The fundamental changes are building blocks for ever more complex structures. At approximately eleven or twelve years of age, children have attained the formal operations period which allows combinatorial analysis a n d propositional thinking. These operations are the foundation f o r adult's hypothetico-deductive reasoning. This development is aptly described in the following quotations. At the level where the hypothetico-deductive operations begin (the ability to reason about a proposition considered as a hypothesis independently of the truth of its content), we see the emergence of a new structure which results from a second form of connection between structures involving inversions and those involving reciprocities (Beth and Piaget 1966: 180). To describe this new structure, we shall for convenience employ the usual notation of the two-valued logic of propositions, but we stress that this in no way implies, either that the subject imposes on himself rules equivalent to the logician's axioms, or that the natural employment of the operations which we shall write (p q), (pvq) etc., conforms to the logician's usage. We have merely stated that at the hypothetico-deductive stage the pre-adolescent or the adolescent no longer restricts himself to reasoning from simple inclusions or from seriations etc. ..., but according to the different possibilities consistent with a combinatorial system, hence according to 16 possibilities for the 4 basic associations (AB, AE, SB, and AB). These 16 possibilities then correspond to the 16 binary operations which it is possible to form with two propositions p and q and their negations. We shall therefore use the ordinary propositional symbolism to designate them, instead of constructing a special symbolism for these 'natural' propositional combinations, but, let me repeat, without making any assumption about the correspondence between formal structures and natural structures, unless they both include the same elementary combinatorial system (Beth and Piaget 1966:180-181). Piaget was aware of the potential confounding of a formal, logical characterization of behavior with the actual problem-solving behavior itself. In much of his work, he maintained that the relationship between the two levels of analysis was one of isomorphism or parallelism, yet the proviso of the last quotation suggests an identity

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relationship if the same operations and combinatorial system are evident in each. And in fact, the identity relationship appears to have been asserted in the following: Insofar as the judgements stated by the subjects correspond to operations in the propositional system, and insofar as these operations can be formulated by means of algebraic symbols (as we have formulated them here), the reasoning of these subjects corresponds to these transformations which link these operations together. No further operations need be introduced since these transformations correspond to the calculus inherent to the algebra of propositional logic. In short, reasoning is nothing more than the propositional calculus itself [emphasis added] (Inhelder and Piaget 1958: 305). The formal model has clearly been advanced here as a cognitive processing strategy. On the one hand, this is not too surprising since much of Piaget's theory of knowledge is based on logico-mathematical structures and their attainment. On the other hand, the presupposition of a logical, mathematical, universal system appears paradoxically at odds with the epigenetic approach. It seems that a truly epigenetic perspective would attempt to describe the rules and relationships onto which a subjects' behavior mapped rather than suppose an a priori logical map to describe problem-solving behavior. The danger in the latter approach is the potential lack of correspondence between the assumed map and the strategies of the individual. This problem is not resolved and many inconsistencies add to the confusion. Of course, the subject does not think about the operations he uses and he could not formulate them. As we insisted under (I), the structures we are discussing here do not exist as distinct concepts in the subjects' consciousness, but are only manifested in his behavior. Thus it is the observer and not the subject who notices and formulates them by referring to a model (Beth and Piaget 1966:181). Now this new fact is fundamental. Up to this stage of development, we could only observe in the subject's behavior, structures limited to the reversibility either of inversions (groupings of classes) or of reciprocity (groupings of relations). With the appearance of the propositional combinatorial system we see, on the contrary, a complex structure elaborated,

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combining in a single system the two types of combinations which thus far were independent [emphasis added] (Beth and Piaget 1966: 182). The paradox is unresolved and we are left with a rather vague conceptualization of people acquiring logico-mathematical structures and the propositional system and applying it to information in propositional operations. The formal model of propositional calculus and the cognitive processes of people appear to be the same. This view is protected from refutation by (a) the presupposition of the final cognitive state and (b) insistence that empirical research cannot test the logical structures. With regard to the first point, Piaget's entire concept of cognitive development presupposes that hypothetico-deductive reasoning is the final product of cognitive development and each preceding stage is in some way helping the child approach it. The presupposition of a propositional logical model dictates both the task selection and the epistemic analysis (e.g. combinatorial analysis). Understanding of cognitive processes here is hampered by trying to fit observable behavior into a preconceived model of the final state. Concerning the second point Piaget said, So, without going back to the classical representatives of psychologism, it is clear that recourse to psychology can lead back to the seductions of empiricism, and for this reason it is appropriate to begin this concluding chapter by recalling why genetic analysis has convinced us that this was a fundamental misunderstanding (Beth and Piaget 1966: 281). Piaget's rejection of empirical research is possibly a rejection of the behaviorist school of thought current twenty or thirty years ago, but the repudiation of experimental analysis serves to protect his model from disconfirmation. Piaget has made many extremely important contributions to genetic epistemology, many of them beginning as tenuous inferences only to be confirmed later by diligent researchers. However, the paucity of empirical support for some of the inferential analyses has often been emphasized (cf. Braine 1968). While reading Piaget's descriptions of children solving tasks, one is struck by the fact that the mathematical and logical formulations may be extraneous and

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unnecessary for the behavioral solution. Smedslund (1966: 165) said, "... Piaget's descriptions of tasks in terms of specific logical structures alone appear clearly inadequate". Indeed, the evidence used to illustrate the sixteen propositional operations has been recently challenged. Bynum, Thomas, and Weitz (1972) pointed out that Inhelder and Piaget's total evidence for the sixteen binary operations of propositional thinking were derived from a single protocol on the physical task — Role of Invisible Magnetism. Their analysis of the original data revealed evidence of only eight of the sixteen operations described by Inhelder and Piaget. The meager empirical support makes one skeptical, at least, of the validity and generality of the presumed propositional operations. In summary form, Piaget proceeded something like this. First, functional, mathematical operations in logic and behavior were identified, and their independence was asserted. (Piaget asserts that mathematical logic has existence independent of the perceiver.) The next step was to show how elements of the formal model were evident in behavior. The task selection was appropriate and logical operations were "demonstrated". Then, the formal model was adopted as the final product and as the process, with all the excess theoretical baggage. Gaps were filled in with hypothetical constructs such as "reflective abstraction". The model is preserved by the idiosyncratic language and constructs of the theory, and is isolated because it was already denied that one could empirically test for the formal model in behavior. So the formal logical model cannot be refuted and it received some support from inferences from primarily one task. The fundamental problem with the approach is that there are no safeguards to preserve mutual interaction between the formal model of processing and the actual cognitive operations employed. What we are left with is a nonbehavioral, reified, formal, logical account of a universal problem-solving strategy and little understanding for the manner in which people at different ages understand information in propositional relationships.6 9

Within the Piagetian model, there has been considerable research demonstrating developmental responsiveness to specific training of propositional logic (cf. Youniss, Furth, and Ross 1971). This area of research is interesting and

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1.5. THE PSYCHOLINGUISTIC APPROACH

A recent approach to the problem of how subjects process propositional relationships recasts the task in a verbal modality. Again, the formal model provides a structural description of the task. Subjects are typically presented with a verbal or written statement of a propositional relationship such as: A and B, A or B, and if A, then B. The language connectives imply the appropriate propositional logical relation in each case between the elements A and B. The tasks are thus structurally identical with nonverbal card-sorting methods as well as with the formal model of propositional logic. A good example of this type of research is provided by the work of Neimark (1970a; 1970b). Her tasks involved (a) directions to subjects to circle all elements in an array described by A or B, (b) matching verbal propositions to circled sets of elements, and (c) writing verbal descriptions of particular element combinations. It was found that comprehension of disjunctive connectives was poor for children from grades 3 through 10 but improved for subjects in grades 11 and 12. Few subjects produced disjunctive descriptions of disjunctively related elements in the array. This contradicts the contention that children as young as six years of age comprehend propositional relationships (cf. Bourne and O'Banion 1971), and it may also contradict the assumption that sentences with "or" are readily understood. Another line of psycholinguistic research has been concerned with the processing of conditional and biconditional sentences. A task often employed by European investigators presents four cards to subjects. Each has a number on one side (odd or even) and a letter on the other (vowel or consonant). Four different sides are exposed and subjects are asked to turn over as few cards as possible to evaluate the truth of a statement such as, "If there is a vowel on one side, then there is an even number on the other." Subjects may also be shown both sides of a card and asked if it makes the rule important with regard to learning abstract classification systems but is somewhat orthogonal to normally applied comprehension processes in the absence of training.

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true or false or as irrelevant. These are often referred to as the selection task and the evaluation task (Legrenzi 1970). The results of studies using such tasks indicate that subjects consistently do not choose the logical choice cards nor do they correctly evaluate the rules from instances. Subjects in the selection task show a marked tendency to choose positive instances although the best choice is a negative instance (since only the TF form can make the conditional rule false). This is remarkably similar to the positive focusing strategy discussed earlier in card-sorting tasks. Similar nonlogical performance is evident in the evaluation task where subjects usually judge FT truth forms as falsifying the conditional rule and FF instances as irrelevant (Wason 1968). The TF and FT instances are consistently judged as false in conditional propositions (Matalon 1962; Peel 1967; Shapiro and O'Brien 1970; Wason 1968). Even when the operations of conditionality are verbally stated in other forms, the traditional material equivalence of the mixed truth forms is obtained. These results suggest that comprehension of language connectives and their implied propositional relationships is poor and not in accord with the formal model of propositional logic. The results of psycholinguistic assessments of processing of propositional relationships seriously limits the generalizability of propositional logical thinking propounded by the concept identification and Piagetian approaches. The formal model simply does not adequately describe how subjects comprehend language connectives and their implied propositional relationships. One might argue that these dissonant results merely reflect different processing modalities. However, there are at least two objections to the postulation of different verbal and nonverbal cognitive processes. First, it maintains the reified concept of propositional logical thinking as an entity and an explanatory behavioral strategy. The circularity of this concept fails to investigate how subjects actually process these relationships in verbal or nonverbal tasks and offers little more then nominal description. The second reason is the unattractiveness of partitioning cognitive abilities to explain different dependent variables. Mehler and Bever (1968) stated the problem succinctly:

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There is no a priori reason to assume that a particular basic structure underlying apparently different behaviors is represented by different internal mechanisms : for instance, we do not need to postulate one kind of structure for the solution of logical problems and another for the perception of linguistic stimuli with formally identical internal structure.

If only one internal mechanism is postulated, though, we are left with two conflicting conclusions regarding subjects' problem-solving capabilities. In the light of the previous discussion of the theoretical and methodological weaknesses of the card-sorting approach it appears imprudent to accept the formal model of propositional thinking. Unfortunately, there is still precious little empirical evidence regarding the comprehension processes applied by subjects to propositional relationships. The psycholinguistic research is valuable because it has been virtually the only alternative experimental methodology employed. It is not without problems, however. The psycholinguistic tasks provide additional linguistic features, and the syntactic and semantic relationships between elements of verbal propositions have not been adequately explored. However, the tasks may be less contrived than nonverbal tasks and more closely aligned to the everyday experiences which we wish to explain. The psycholinguistic approach, therefore, is heuristically valuable for the understanding of language processing and methodologically promising as the only experimental alternative advanced to investigate comprehension of propositional logical relationships.

1.6. SUMMARY

The experimental analysis of how people process propositional logical relations has traditionally relied on nonverbal card-sorting tasks. However, the tasks possess serious methodological problems which weaken their internal validity and the inferences drawn from them. Theoretically, one must argue against the equivalence of paired-associate learning with abstract knowledge of propositional rules. The reification of propositional logical thinking

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as an explanatory account, in the guise of intuitive truth table strategies or formal operations, offers little to our understanding of subjects' problem-solving behavior on these tasks. The psycholinguistic tasks provide alternative and additional assessments of the manner in which propositional relationships are comprehended. Future research in this area must not be unduly influenced by nominal, formal descriptions of the task but rather focus on specific problem-solving strategies such as positive focusing, reasoning from negative instances, memory requirements, and semantic relationships among constituent elements. These can be common grounds for research with both verbal and nonverbal tasks. This approach avoids the unparsimonious partitioning of mental faculties and the reification of formal models as explanations of cognitive processes.

2 EMPIRICAL RESEARCH

2.1. RATIONALE Both Piagetian theory and concept-identification research assert that adolescents and children attain the cognitive abilities to comprehend and utilize propositional rules. However, the research using verbal assessment methods does not corroborate this conclusion. The present research is an investigation of this contradiction using psycholinguistic tasks and may be considered a derived test implication from Piagetian theory and concept-identification, if one is willing to assume that general cognitive abilities should be equally evident in verbal as well as nonverbal tasks. At a different level, the differences in tasks can be viewed as a test of the generality of the hypothesized capacities. If propositional rules, are, in fact, general problem-solving strategies of people, then they should be evident in verbal tasks which require the same operations for solution. The new tasks provide certain linguistic idiosyncrasies which seem profitable. The methods are free of complicated pretraining, exemplars, the complexity of the stimulus dimensions, and attributes of card sorting, and appear to be less artificial and contrived. The advantages of this type of assessment are thus both theoretical and methodological. Probably the best justification for the present research is the virtual void of information concerning comprehension of language connectives (see Anisfeld 1968; Bar-Hillel and Eifermann 1970). The little research done on comprehension of logical relations in language with adults and children has often been isolated from the

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mainstream of psychological research and theory. Therefore, the provision of heretofore unavailable diagnostic information and its relationship to extant theories is another primary goal of this investigation. But most important, this study is an attempt to understand how children and adults understand verbal-propositions. A major focus is the analysis of actual problem-solving strategies of subjects without the a priori assumptions of a formal logical model. The goal is to understand subjects' cognitive processes on these tasks and their integration with other cognitive abilities.

2.2. E X P E R I M E N T I

There are three important dimensions of this study which deserve attention. Experiment I examined comprehension of eight different language connectives, each one implicitly asserting a propositional logical relationship between the constituent elements of the sentence. With regard to other psycholinguistic research, this is an extension in the number of rule relationships studied as well as the number of connectives. A second factor of Experiment I was the truth-form representations. Each logical rule was expressed in each of the truth forms for each connective. In this manner the task was directly analogous to nonverbal card-sorting tasks. The third factor was age. Experiment I was cross-sectional in design, drawing subjects ranging from seven to twenty-two years of age in order to test developmental changes in comprehension of logical relationships in language. As a consequence of this design, extended and repeated observations on each subject were not possible.

2.2.1.

Hypotheses

1. Performance was expected to improve with age. This is a direct implication of Piagetian theory. Further, one might presume that increasing language skills and decreasing test-taking errors would contribute to the improvement.

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45

2. Comprehension differences between connectives were anticipated. Conjunctive connectives were expected to be the easiest, the disjunctive somewhat more difficult. On the basis of the psycholinguistic research and unfamiliar logical relationships, conditional and biconditional sentences were expected to be difficult. These anticipations were for subjects of all ages. 3. Similarly, all subjects were expected to find the mixed truth forms (TF and FT) more difficult to comprehend than the homogeneous pairs for all connectives. The reasons for this hypothesis are the "biasing habits" of card sorting and preference for conjunctive (TT) relationships. 2.2.2. Method Subjects. Children in grades 2, 5, 8, and 11 attending public schools in Bedford, Indiana, participated in this experiment. Although three different schools were represented in this sample, all the students involved were from the same geographical area and similar socioeconomic status. All subjects were native speakers of English. Three intact classes at each grade level were given the groupadministered test. Additionally, three groups of college students from the introductory psychology courses' subject pool participated. All subjects' data was not pooled and further information regarding the final sample is discussed later. Task. The task was a diagnostic assessment of the individual's ability to comprehend logical relationships expressed in verbal, propositional forms. The task was a True-False, paper-and-pencil test in which subjects' judgements were dependent upon the correspondence between projected pictures and the verbal propositions. Each trial consisted of a slide presentation for fifteen to twenty seconds and the simultaneous reading of a descriptive proposition. The subjects' task was to decide if the description was true or false. Altogether there were thirty-two such test trials and eight control trials (see Appendix A). Total time of the task was approximately twenty minutes. The thirty-two test sentences were derived from the matrix of

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two factors, eight linguistic connectives, and the four forms of the logical truth table. The following connectives were used to generate the compound test sentences: AND, BUT, BOTH—AND, NEITHER— NOR, EITHER—OR, IF—THEN, and IF AND ONLY IF—THEN. These connectives exemplify the logical relationships of conjunction (conjunctive absence in the case of NEITHER—NOR), disjunction, conditionality, and biconditionality. Table 2 illustrates these relationships for all thirty-two test item representations. TABLE 2 .

Linguistic Connective

and but both—and neither—nor or either—or if—then if and only if—then

The Logical Truth Table Representations of the Linguistic Connectives Logical Relationship

Conjunction Conjunction Conjunction Conjunctive Absence Disjunction (Inclusive) Disjunction (Inclusive) Conditionality Biconditionality

Truth Forms TT

TF

FT

FF

T T T F T T

F F F F T T F F

F F F F T T T F

F F F T F F T T

T

T

The sentences and slides used in this task were based on eight pairs of pictures included in the Sequential Picture Cards II kit (Developmental Learning Materials, 3505 North Ashland Avenue, Chicago, Illinois, 60657). The pairs of pictures were related in an obvious fashion. For example, one member of a pair pictured a boy riding a bicycle and the other member pictured the boy standing next to the bicycle. One member of each pair was designated the positive or true instance of the pair and the other the negative or false instance. Eight simple, active, affirmative, declarative, present-tense sentences were generated which described the positive instances. For each connective, there were four truth forms. Each form contained two component or atomic sentences, thus a total of eight component sentences per connective. Each of the eight positive instance sentences was paired with one of the other seven sentences

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47

to generate an exhaustive pairing of the component sentences with the four truth forms of each connective. Among the eight connectives, each positive instance sentence was paired with each of the other seven at least once. Although there were no negative sentences to describe the negative instances, a logically false relationship existed when a positive instance sentence was read with a negative instance slide presentation. Thus, each component positive instance sentence appeared eight times in the test sentences but was a true description of the slide presentation only half the time; each member of each picture pair appeared four times among the thirty-two test items. Because of the logical relationships exemplified in the connectives, there were seventeen logically false test sentences and fifteen logically true ones. The additional eight sentences in the task were simply the component, positive-instance sentences used as a control to insure comprehension of the individual components of the compound test sentences. These sentences also served as controls for errors due to factors such as losing one's place on the answer sheet. Half of these control items appeared with the positive instance pictures (T) and half with the negative instances (F). The actual forty-item test list was constructed by randomly assigning the numbers 1 through 40 to each slide-sentence pair. Although the list was randomized initially, it was presented in the same order to all groups. The answer sheet was stenciled and contained forty pairs of response alternatives, the typed letters 7" and F. The subjects were instructed to cross out or X the appropriate letter. The 35 mm. color slides of the positive and negative pictures were displayed via a Kodak Carousel projector to make a projected image approximately four to five feet square. Procedure. The male experimenter was introduced to each classroom by the teacher and stated that he was interested in the way people understand different kinds of sentences. The answer sheets were passed out and instructions were given concerning the name, age, and sex blanks to be filled in. The instructions included a brief explanation of a True-False test and two simple-sentence verbal examples. It was explained that a picture would appear on the screen

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and then a sentence would be read. The subjects' job was to decide if the sentence was TRUE or FALSE in relation to the picture. The instructions were designed to ensure attention to the sentences and an understanding of the task requirements. Each slide appeared for fifteen to twenty seconds during which the sentence was read twice in as uniform manner as possible. Measures. Several potential problems exist with the group design of this experiment. If one were to count errors of comprehension on all connectives, there is the danger of including subjects who missed the items for reasons other than logical relationship comprehension. The control items eliminated several possibilities, for example, misunderstanding of the component or simple sentences. Other possibilities include deliberate errors or losing one's place on the answer sheet. Therefore, any subject committing an error on any of the eight control items was dropped from subsequent data analyses. It is recognized that this does not eliminate all errors due to factors other than comprehension of the constituent relationships nor necessarily all of those subjects committing the above possible errors. However, it does seem a judicious safeguard. The attrition at grades 2, 5, 8,11, and college were 26 percent, 8 percent, 11 percent, 7 percent, and 6 percent, respectively. The resultant sample sizes, in the same order, were sixty-three, seventy-nine, sixty-seven, seventy-one, and fifty-nine. Another problem exists, however, in administering a task to an intact group and that is the nonindependence of individual subjects. It is entirely possible that an idiosyncrasy of the testing procedures or the specific groups themselves could lead to significant differences between groups within the same age level. To test the homogeneity of errors among groups at each level grade, a two-way analysis of variance, Groups (three) X Connectives (eight) was computed using the total number of errors for each connective for each subject as the dependent variable. This comparison of mean errors resulted in nonsignificant main effects and interactions for all grades except grade 8. Grade 8 showed a significant group effect, F(2,66) = 6.73, p < .01, and interaction, F(14,462) = 1.71, p < .05. It was later determined that one of these classes contained students character-

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49

ized as slow learners and behavioral problems. When this class was excluded from the analysis, there was no significant effect for groups or interaction. The resultant sample size of the two remaining eighth-grade groups was fifty-one with an attrition of 10 percent. 1 On the assumption of homogeneity among groups, a sample of twenty male and twenty female subjects was randomly drawn from the total data at each grade level. The mean chronological age for each of the grade levels in ascending order was 7 years 9 months, 11 years 0 months, 13 years 8 months, 16 years 7 months, and 19 years 2 months. These 200 subjects comprised the data for subsequent analyses. 2.2.3. Results and Discussion One might ask if there was a response bias on the True-False test. Although there were nineteen TRUE test items and twenty-one FALSE items, there was actually a greater imbalance in the data. Subjects gave consistently more FALSE responses. The mean number of FALSE responses per group in ascending age order was 24.4, 25.6, 24.6, 24.4, and 24.0 — no differences in response bias among grade levels. This outcome suggests that: (a) most errors were due to responding FALSE to logically true propositions, and (b) the pattern was fairly stable across grades. This does not imply that the number of errors was constant across grades; only that most of the errors were due to inaccurately responding FALSE. It was hypothesized that performance would improve with age on this task. The random test-taking errors were eliminated as a source of bias and any resulting differences should reflect actual differences in cognitive abilities. Table 3 illustrates the errors on each connective for each grade collapsed across sexes which was a nonsignificant factor. As the right hand margin indicates, there was a steady reduction in total errors until grade 11. Each step shows a decrease of roughly 1

In addition, it should be noted that all five grade levels showed reliable differences among connectives (ps < .001).

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With striking consistency, however, twenty-one second graders responded TRUE to TT sentences and FALSE to F F sentences involving NEITHER—NOR. This is an important observation because it suggests that many of the second graders were constrained by features of the constituent elements regardless of the language connective conjoining them. It should also be noted that performance on the first three connectives in Table 6 would lead one to conclude that second graders had the ability to perform conjunctive operations. Their many errors on NEITHER—NOR sentences suggest that negation could not be incorporated into that operation or that the sentences were all answered on a basis other than logical conjunctive operations. Disjunctive Connectives. There are two kinds of logical disjunction, inclusive and exclusive, and the only difference is in the treatment of the TT form. Inclusive disjunction accepts as true any proposition where at least one constituent is true. Exclusive disjunction allows true propositions to contain one and only one true constituent. Both disjunctive connectives, OR and EITHER—OR, were scored as inclusive disjunctions. The reason for this was the ambiguity of the correspondence between the verbal and logical relationships. No other connective could be ambiguously interpreted according to the formal logical model. Of the 400 TT disjunctive propositions across all subjects there were only eighty-three FALSE responses. In other words, only 21 percent of all responses could be presumed to be correct answers to TT exclusive disjunctive relationships. The disjunctive connectives were scored as inclusive disjunctions because the majority of subjects responded to them in that manner, and because the nominal designation of an error is unimportant for the identification of the processing strategies according to which subjects comprehended the connectives. The tendency to treat disjunction exclusively was more pronounced among the older subjects and is visible in Table 7. One can ask how much of the total errors on disjunctive propositions can be accounted for by the TT form in order to assess the relative amounts of exclusive disjunctive processing. Only 4 percent of second graders' errors on disjunction was due to responding FALSE

56

EMPIRICAL RESEARCH TABLE

7.

Percent Errors on Disjunctive Connectives According to Truth Form and Grade Connectives

Or

Either-or

Grade

TT

TF

FT

FF

TT

TF

FT

FF

2 5 8 11 C Total

2.5 22.5 15.0 25.0 25.0 18.0

67.5 70.0 50.0 30.0 27.5 49.0

72.5 62.5 47.5 27.5 20.0 46.0

2.5 2.5 2.5 .0 .0 1.5

7.5 25.0 25.0 27.5 32.5 23.5

50.0 57.5 35.0 10.0 12.5 33.0

62.5 45.0 37.5 10.0 10.0 33.0

5.0 17.5 10.0 .0 .0 6.5

to TT items. On the other hand, 45 percent of college subjects' errors reflect possible exclusive disjunctive processing of the TT propositions. This does not appear to be an artifact of overall age-related reductions in errors. The percent of errors on TT disjunctive items relative to total disjunctive errors increased steadily from the younger to the college subjects — 4 percent, 16 percent, 18 percent, 40 percent, and 45 percent — as do the total errors on TT disjunctive items — four, nineteen, sixteen, twenty-one, twenty-three. Returning to Table 7, though, there are several interesting contrasts which are significant — most obviously, the greater difficulty of the mixed truth forms. For example, the mixed forms for the second graders showed performance at or below chance levels. This stands in sharp contrast to their excellent performance on TT and FF forms. It is evident, however, that the difficulty of TF and FT items diminishes with age. The systematic improvement on these forms suggests the acquisition of a more efficient processing strategy. Surprisingly, though, adults had some difficulty with mixed form disjunctive sentences. Despite improvement with age, disjunctive propositions were always more difficult to evaluate than conjunctive propositions. There is a tendency for performance to be superior for EITHER— OR propositions relative to OR sentences, ^(1,190) = 7.83, p < .001. The improvement is virtually all in the mixed truth forms. Similarly

EMPIRICAL RESEARCH TABLE

57

8. Percent Errors on the Conditional and Biconditional Connectives According to Truth Form and Grade i f --then

if and only if—then

Grade

TT

TF

FT

FF

TT

TF

FT

FF

2 5 8 11 C

0 0 5.0 7.5 17.5

17.5 2.5 .0 .0 2.5

75.0 95.0 100.0 100.0 95.0

92.5 72.5 42.5 40.0 45.0

10.0 5.0 15.0 10.0 20.0

5.0 7.5 2.5 2.5 .0

17.5 7.5 10.0 .0 .0

90.0 72.5 50.0 42.5 50.0

Total

6.0

4.5

93.0

58.5

12.0

3.5

7.5

61.0

more exclusive processing errors were obtained on this connective. It seems reasonable to speculate that the distributive quantifier EITHER introduced an additional cue to differentiate the constituents which facilitated correct processing of TF and FT items and promoted exclusive processing of disjunctions. Conditional Connective. Table 8 shows the error patterns associated with each truth form and grade for IF—THEN sentences. The overwhelming feature of this table is the extreme difficulty of the FT and FF items, both of which require a TRUE response. These two columns show interesting performance changes with age, though. Except for the second graders, the errors on FT items are near or at the ceiling level. On the other hand, errors on FF items decrease from ceiling to near chance levels. The errors suggest that subjects consistently employed erroneous, nonlogical processing strategies. Also of interest is the modest tendency for errors to increase with age on TT items. Overall, these factors cancel each other and there are no age-related differences. Biconditional Connective. The errors made by all grades on the IF AND ONLY IF—THEN items are also shown in Table 8. The FF truth forms account for the majority of errors. Again, as with conditional propositions, a trend for above chance errors declining to chance levels with increasing age is visible for the FF items. In fact, there is remarkable similarity in the performance on conditional

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and biconditional propositions. The errors tend to increase slightly with age on TT sentences, while TF items are handled nearly perfectly with both connectives. The conditional FT items received fourteen TRUE responses which were correct while the biconditional F T items received fifteen TRUE responses which were errors. By and large, it appears that both connectives were processed in the same manner and only the logical definition of an error changed. Strategy Analysis. Many of the subjects in Experiment I appeared to ignore the logical relationship implied by the connectives and responded according to a type of mismatch strategy. In other words, whenever any features of the sentence and picture did not correspond, subjects responded FALSE. This is similar to a conjunctive set or matching to sample for the TT items. The pattern of responses predicted by the mismatch strategy is TRUE for TT items and FALSE for TF, FT, and FF propositions. The thirty-two actual test sentences yield nine characteristic errors if this strategy is applied perfectly. In other words, a subject following this algorithm would err on the TT and FF NEITHER—NOR sentences, the TF and FT OR and EITHER—OR sentences, the FT conditional sentences, and the FF conditional and biconditional items. A reinspection of these cells in Tables 6, 7, and 8 reveals that all nine of the error rates for the second graders were above chance. The magnitude of errors within these cells decreased with age generally but persisted as the most difficult items. Another way of approaching the issue is to inquire as to how many of the subjects' responses corresponded to the predicted responses of the mismatch strategy. This information is summarized in Table 9. The 92+ percent entries in the TT and FF cells for second graders are strong support for the inefficient mismatch strategy. The data for the mixed truth forms and older subjects is not as compelling as that for the youngest grade tested, though. It should be pointed out that while reading this table, comparable percentages of responses do not necessarily imply comparable error rates, as evident when directly compared to Table 3. Another approach to the strategy-identification problem is to analyze the individual protocols. The difficulty here is the selection

EMPIRICAL RESEARCH TABLE

9.

59

Percent Total Responses Conforming to Mismatch Strategy Truth Form TT

TF

FT

FF

Predicted Response by Mismatch Strategy Percent Correct if strategy followed perfectly

T

F

F

F

87.5

75.0

62.5

62.5

Grade 2 5 8 11 C

92.5 80.3 80.6 76.3 75.0

76.3 86.6 84.4 78.8 79.1

80.0 83.8 82.8 78.8 77.2

92.2 79.4 72.5 73.1 74.1

Total

80.9

81.0

80.5

78.3

of an appropriate criterion by which to judge if a subject did in fact follow a particular strategy. The most stringent requirement is to count those subjects who missed all nine and only those nine items predicted by the mismatch strategy. A less restrictive criterion would be counting those subjects who missed seven or more of those nine propositions, thus allowing for chance deviations from the strategy. Even this criterion may be inappropriate, however, because the NEITHER—NOR TT and FF items were only difficult for the second graders. Regarding only the other seven sentences as critical indicators of the mismatch strategy, two similar criteria can be advanced — seven of seven errors and five of seven errors. These four criteria are all shown in Table 10 along with the number of subjects meeting the criteria at each grade. The multiple-criterion analysis is necessitated by the lack of a single best criterion and by the more complete information provided. Six of the forty second graders committed all nine and only those nine errors predicted by the mismatch strategy, while thirteen others missed at least seven of the nine. Again, there is strong evidence to support the mismatch strategy as an individual information-processing rule for the second graders. It thus appears that many of these children were responding

60

EMPIRICAL RESEARCH TABLE 10. Number of Subjects in Each Group Following Particular Error Patterns Strategy With neither-nor connective

Criteria Without neither-nor connective

Error Patterns Grade 2 5 8 11 C

7/9 19 8 5 1 1

9/9 6 0 0 0 0

5/7 28 24 17 4 7

7/7 1 5 2 1 1

solely on the basis of the feature's presence and absence regardless of the logical relationship implied by the language connective. To a lesser extent, and with improvement on NEITHER—NOR, the fifth graders showed similar patterns. As with Table 9, Table 10 suggests that few older subjects were constrained to such an extent by the stimuli. A variety of evidence from diverse sources suggests that the mismatch strategy is a potentially good interpretation of the subject's cognitive processes. For example, the mismatch strategy resembles the stimulus bound strategy described by Youniss, Furth, and Ross (1971) for their Stage 1 subjects who were taught, but could not learn, the symbolic relationships of conjunction and disjunction. Those subjects were also highly constrained by the presence and absence of the stimulus features rather than the logical relationship between them. A similar process was suggested by Bourne (1970) to account possibly for a subject's card-sorting behavior. Referring to Hovland (1952), Bourne suggested that subjects could focus on positive instances (TT), assigning them to the positive category, and assign the remaining instances to the negative category by default. A congruent strategy, in outcome at least, is the predisposition or "biasing habits" to process two-featured information as conjunctive relationships (Bourne and O'Banion 1971; Bruner, Goodnow, and Austin 1956).

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It should be pointed out that this strategy is presumed to occur across connectives and logical relationships. It is therefore a general strategy which could be applied with differential efficiency to particular connectives. Also, it is possible that the mismatch strategy interacts with other strategies and does not operate in isolation. This is, perhaps, more plausible and will be discussed later. 2.2.4.

Summary

The results of Experiment I indicated that children and adults had considerable difficulty in comprehending abstract logical relationships expressed in verbal propositions. On the basis of extant models of cognitive processes, as exemplified by Bourne and Piaget, one would not expect this difficulty. The main thrust of this data is rejection of the logical processing model to extensions of language comprehension, and a mandate for detailed analysis of the actual problem-solving strategies employed by subjects of different ages. In addition to rejection of an ideal processing model, the data provides important diagnostic information. The improvement with age in comprehension on most, but not all, test items suggests similarities and dissimilarities in age-related strategies. The relative difficulty of disjunction and conditionality may be surprising but there has been little research in this area of psycholinguistics. Many possible factors arise as bases for this difficulty. For example, the connectives may have to conjoin related constituents and the type of relationship may be crucial. Possibly disjunction is easily understood if the constituents are exclusive or exhaustive. In any event, a variety of interesting parameters are suggested which appear more closely aligned with typical experience and problem-solving behavior than the abstract model of propositional calculus.

2.3. EXPERIMENT II

The results of Experiment I suggested that the formal model of propositional logic did not adequately characterize subjects' com-

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prehension of language connectives. This forced us to consider alternative explanations. The mismatch strategy was proposed as a possible information-processing strategy because its predicted errors were similar to those obtained in Experiment I. It was a viable alternative because it was not couched in formalistic models but empirically derived according to the error patterns. Furthermore, it was proposed as a general strategy applicable to all connectives and subjects. The purpose of Experiment II was to further investigate the viability of the mismatch strategy. If one considers the mismatch strategy to be serial and self-terminating, it predicts that subjects cease attending if the first element is false. It is possible that subjects are capable of comprehending the relationship but do not because this strategy leads to inattention to all the stimulus information. The test implication derived from this hypothesis suggests that subjects' response times can be measured to determine if premature responding occurs. Experiment II employed a task similar to Experiment I but reaction times (RT's) as well as errors were collected for each proposition. In this manner, it could be observed whether subjects intruded their responses during the sentence presentation. A primary feature of Experiment II was the replication of Experiment I with similar pictures and sentences and the connectives AND, OR, and IF—THEN. However, there were several advantages to the present study. Each type of proposition had two examples, thus allowing a measure of subjects' response consistency which was not possible in Experiment I. Further, there were no subjects dropped from the data analysis in Experiment II, thus overcoming the attrition problem of the first study. Finally, the test was not group-administered, but rather each subject was seen individually. 2.3.1. Method Subjects. Ten subjects each from grades 2, 5, 8, and 11 attending public schools in Bloomington, Indiana, and ten college students from the psychology subject pool participated in this study. Each group was composed of five males and five females. An effort was

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63

made to obtain average, or representative, subjects from the public schools. For example, the eighth- and eleventh-grade subjects were drawn from average English classes, a course taken by all students. The mean ages of the groups in ascending order was 8 years 8 months, 10 years 8 months, 13 years 7 months, 16 years 6 months, and 19 years 4 months. Task. The experimental task was similar to that used in Experiment I. Slide projected pictures were shown, sentences read aloud, and subjects required to respond "True" or "False". Only three connectives were used in this study, AND, OR, and IF—THEN, representing the logical relationships of conjunction, inclusive disjunction, and conditionality. Two sentences of each truth form for each connective were presented for a total of twenty-four propositions. Eight simple component sentences were presented as controls again, half true and half false, yielding a total test of thirty-two items (see Appendix B). The pictures were the same as in Experiment I but the picture-sentence pairs were different. It was in this sense a replication of the first experiment. Apparatus. The color slides were presented via a Kodak Carousel projector as in Experiment I. All sentences were presented via a Norelco tape recorder. The onset of a sentence activated a voiceoperated relay which drove a timer. Depression of either response button, labeled TRUE or FALSE, terminated the timer and activated a small light. The buttons, speaker, and timer were housed in a black wooden box 20" x 8" x 8" and the buttons were 12" apart on the top of the box. Procedure. Each subject was tested individually in a quiet room. The experimenter explained that he was interested in the way people understand various kinds of sentences and the task involved measuring the subject's understanding of different sentences. It was explained that a picture would be shown via the slide projector and a sentence would come on the tape recorder. The subjects' job was to decide if the relationship between the picture and sentence was true or false. The buttons were pointed out and demonstrated with a verbal example of each. The experimenter then read aloud the labels TRUE and FALSE to the second graders. The experimenter showed

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the clock to the subject and explained that it was important not only to get the right answer, but also to do it as fast as possible. It was stressed that the subject should be fast and accurate and that a response could be made anytime after the sentence started. Three single pictures and simple sentences followed as a warm-up to insure understanding of the task requirements. The testing followed immediately. Questions were answered by reiteration of pertinent parts of the instructions. One exception was subjects' query regarding the pictures if "half was right and half wasn't". The standard reply was that the whole sentence was true or else the whole sentence was false and that there was a correct answer for every sentence. Measures. Two measures were recorded for every proposition — the RT to the nearest tenth of a second and the subject's response of TRUE or FALSE. The error scores were tabulated but the RT's were transformed because of the variable times taken to read each sentence. Therefore, the actual time of the sentence on the tape recorder was subtracted from the RT to yield a differential reaction time (DRT). In this manner, the DRT takes into account the variability among sentence lengths and provides a standard measure. A negative DRT indicates that the subject responded before the end of the sentence. If indeed subjects do not attend to the complete sentence and intrude their responses during its presentation, then one would expect to obtain many negative DRT's. 2.3.2.

Results and Discussion

Error Scores. The raw data was summed over the two examples of each proposition and over sexes to yield the information displayed in Table 11. Inspection of the extreme right-hand margin reveals a modest decrease in errors with increasing age. It is also clear that the three connectives varied in difficulty. Conditionality was the most difficult relationship while conjunction was the easiest. However, as in Experiment I, comprehension of the proposition was dependent upon the particular truth form of the expression. The main effects due to connectives and truth forms and their inter-

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