The Development of Transitive Inference: A Review of ... - PsycNET

Psychological Bulletin 1978, Vol. 85, No. 6, 1327-1343

The Development of Transitive Inference: A Review of Recent Approaches Elizabeth S. Thayer. and Charles E. Collyer University of Rhode Island Research on the development of transitive inference in children is reviewed, and two principal approaches, one associated with Piaget's stage theory, the other associated with an information-processing theory, are compared. The two approaches differ sharply with respect to four aspects of method: choice of task, response required, initial training, and method of feedback. The divergent conclusions of the two approaches regarding the age at which transitive inference emerges are discussed in relation to these points of methodological divergence. Several issues in the area of transitive inference are viewed as prototypical of diagnostic problems encountered in cognitive development generally.

When normal adults are told that (a) Jack psychologists and has generated a variety of is taller than Jeff and (b) Jeff is taller than approaches and theoretical ideas. The purMark, it is usually possible for them to infer pose of this review is to examine these apthat Jack is taller than Mark. This example proaches, with particular emphasis on a illustrates the type of reasoning called transi- divergence in methodology and conclusions tive inference. More generally, transitive in- between two principal schools of thought in ference is the type of reasoning in which a this area. In addition to providing a sumrelation (e.g., taller than) between a first and mary of these approaches, the review notes a third term can be inferred, given that this a number of aspects of the research on transirelation exists between the first and second tive inference that seem to represent very and between the second and third terms. Of general problems of cognitive-developmental course, transitive relations may exist for sets diagnosis. of more than three terms, and transitive chains of terms are of great importance both Two Approaches to Transitivity in abstract disciplines such as logic (cf. Jean Piaget and others have proposed that Langer, 1967) and in everyday life. children are unable to perform transitive One is confident that most adults are capinferences until they pass from the stage of able of successful transitive inference. The preoperational thought to the stage of confollowing question regarding children, howcrete operations at approximately age 7 ever, seems to be basic: When and how does transitive inference emerge in the course of (Flavell, 1963; Piaget, Inhelder, & Szemincognitive development? This general ques- ska, 19,60). Indeed, among Piagetian develtion has been investigated by a number of opmental psychologists, the transitive inference task is one of the most frequently employed measures of the development of concrete operation. Piaget and his colleagues The authors thank Janet Kulberg for valuable discussions concerning this article, and an anony- propose that young children (below age 7) mous reviewer for helpful criticism of an earlier cannot combine separate experiences to prodraft. The second author's participation was facili- duce a new inferential solution. In particular, tated by a Faculty Fellowship awarded by the Uni- given the basic premises that A is longer than versity of Rhode Island. Requests for reprints should be sent to Elizabeth B (A > B) and B > C, the young child canS. Thayer, Psychology Department, University of not infer that A > C. Piaget believes that Rhode Island, Kingston, Rhode Island 02881. the preoperational child is dominated by Copyright 1978 by the American Psychological Association, Inc. 0033-2909/78/8506-1327$00.75

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immediate perceptual input and cannot reorganize this input once new perceptual items arrive. Perceptual domination and the inability to understand the reversibility of ordered relations prevent the child from making inferences and so constrain logical thinking (Bryant, 1973). Piaget discussed transitivity of length relations within the context of spontaneous measurement in his study of geometrical conceptions (Piaget et al., 1960). The development of this skill comes late in the growth of logical relations. Piaget believes that a child can recognize the lengths of individual terms and the relation between two terms in one pair without being able to order or seriate the lengths of more than two terms. Transitivity enters a child's relational structuring when reversibility and passage among three terms can be understood. At this point in development a particular term in a series (B in the given example) is no longer viewed as having a one-way relation (either smaller than or larger than), but is viewed instead as capable of having two relations simultaneously (both smaller than some things and larger than others). The child's task in the first study directly relevant to transitivity of length concerned the development of certain measuring techniques that presuppose transitivity (Piaget et al., 1960). Children were asked to build a tower of blocks equal in measured height to a tower already built by the experimenter. The experimenter's tower stood on a table that was higher than the table on which the child was to build his/her tower. Sticks of various lengths were made available, but the child was not told how to use them. The criterion of mastery in this task was the child's ability to use the sticks as intermediate comparative measures of the towers. The majority of children appeared to understand and master the measuring technique only after age 7 or 8 years. The principal developmental stages were (a) a crude visual comparison without taking into account the difference in table heights, (b) attempts to bring the towers together for closer visual comparison and the child's use of his/her own body as a common measure, (c) use of body-independent measures such as a tower

or stick exactly the same length as the tower to be measured, and (d) use of sticks longer and shorter than the model. Piaget's assumptions and findings concerning the transitive inference task have been challenged by several investigators (Braine, 1959; Bryant, 1973, 1974; Bryant & Kopytynska, 1976; Bryant & Trabasso, 1971; De Boysson-Bardies & O'Regan, 1973; Harris & Bassett, 1975; Lutkus & Trabasso, 1974; Riley, 1975, 1976; Riley & Trabasso, 1974; Roodin & Gruen, 1970; Trabasso, 1975; Trabasso, Riley, & Wilson, 1975). These authors claim that Piaget's inference task does not, in fact, test whether the young child has inferential ability or not. Their basic hypothesis is that children's differences in making transitive inferences may be more a matter of the memory processes involved than a lack of logical competence. In other words, one must ensure that the child has remembered the comparisons that have to be combined (A> B and B > C) if one is to conclude that the child is or is not making inferences. If memory fails to be controlled, an erroneous response may have little to do with inferential ability or level of logical thinking. Piaget typically did not control for memory factors. Although he has been criticized for this omission by some recent workers, others (Flavell & Wohlwill, 1969; Smedslund, 1963) have argued that the effect of memory training would be to produce only the semblance of a true operation. Their view is that the ability to utilize transitivity acquired through memory training would be short-lived and highly specific to the tasks on which the subject was trained. The memory training procedures used for transitivity tasks have also been said to focus too narrowly on the psychological processes by which information is encoded and decoded rather than on cognitive structures (Flavell & Wohlwill, 1969). This controversy stems from different theoretical assumptions concerning the nature of cognitive development. Piaget and his colleagues conceive of cognitive development as proceeding in a discontinuous fashion, with qualitatively different stages emerging during successive time periods. Each stage represents a unified set of cognitive

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structures; development is not viewed as the continuous maturation of independent psychological entities. Consequently, developmental stage transitions are not denned in terms of purely quantitative changes (Flavell, 1963, 1977). The position taken by Trabasso (1975) is precisely the opposite: "Cognitive development may be seen as continuous and qualitatively similar to the target adult-model. The growth of a child's capacity for immediate, short-term and longterm memory is likely to be gradual and quantitative" (pp. 136-137). In his view of cognitive growth, Trabasso makes a distinction between the skills possessed by the young child and by the adult only on the basis of accumulation of information and the development of more efficient and adaptive methods for processing information. Adult thinking can then be conceptualized as qualitatively similar to that of the young child, but having a larger fund of knowledge, more refined and better-practiced skills, and thus a greater ability to handle cognitively complex problems. Some of the issues addressed in the recent literature on transitive inference offer excellent examples of general problems in cognitive-developmental diagnosis (Flavell, 1977; Smedslund, 1953, 1963). We believe that it is instructive to view these issues within the framework of a distinction made by Flavell (1977) between two sets of questions one may ask about any cognitive acquisition, such as transitive inference. Conceptualization questions ask about the cognitive processes that actually comprise acts labeled transitive inference. These questions are concerned with the kinds of strategies utilized by individuals as they solve any type of transitive inference task. Practical assessment questions, on the other hand, ask what procedures should be used to test for the presence or absence of transitive inference in a child. These questions focus on the methodology and analysis used in the design of a transitive inference task. As one might expect, the assessment procedures deemed appropriate by any investigator depend to some extent on the hypotheses he or she entertains concerning the cognitive processes involved.

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Practical Assessment If we make the assumption that a child either does or does not possess the ability to make a successful transitive inference, then two possible errors can result from an assessment of his or her performance on a specific task. If the child actually does have the logical ability to make a transitive inference and yet shows no evidence of this ability on the task, a false-negative (Type II) diagnostic error will be made, whereas if the child does not possess transitive inference skills and yet somehow makes the correct "symptom response", then a false-positive (Type I) diagnostic error will be made (Flavell, 1977). Both Flavell (1977) and Smedslund (1963) offer a number of potential reasons for committing Type I and Type II errors. False-negative errors can result from (a) failure to understand the initial instructions given by the experimenter, (b) failure to perceive that indeed A is longer than B and that B is longer than C, or (c) failure to remember the two premises (A > B and B > C) long enough to make the transitive inference. The third potential source of error is precisely the area of inquiry pursued by Bryant and Trabasso (1971), In any of these three cases, the tester will misinterpret the absence of the symptom response as the absence of transitivity. Possible sources of false-positive errors include (a) guessing by the child, (b) direct perception (in some tasks) of the A > C relation without transitive inference, or (c) the use of another (nontransitive inference) solution strategy. One such strategy (Smedslund, 1963) would consist of coding the initial premises (A > B, B > C) as follows: "A is long, but B is not long, and B is long, but C is not long." Since A has been coded as being long and C as not long, the child will choose A as longer than C, without making use of the transitive relation among the terms. In any of these cases, the tester will misinterpret the presence of the symptom response as the presence of true transitive inference. Much of the literature on transitive inference has focused on attempting to control for some of these causes of false-negative and false-positive errors. Let us now examine in

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more depth specific research on the variables affecting the acquisition and/or utilization of transitivity. The major controversy in this literature stems from the apparently discrepant findings concerning the age of emergence of transitivity. One group of researchers claims that very young children (age 4 and 5 years) are capable of making correct transitive inferences (Braine, 1959; Brainerd, 1973b; Bryant, 1973, 1974; Bryant & Kopytynska, 1976; Bryant & Trabasso, 1971; DeBoysson-Bardies & O'Regan, 1973; Roodin & Gruen, 1970; Trabasso, 1975; Trabasso, Riley, & Wilson, 1975), whereas another group of researchers supports the Piagetian claim that the age of emergence is around 7-8 years (McManis, 1969; Murray 6 Youniss, 1968; Smedslund, 1960, 1963; Youniss & Furth, 1973; Youniss & Murray, 1970). For the purposes of review it will be useful to examine these studies with respect to four procedural choice points: (a) choice of task, (b) response required, (c) initial training, and (d) method of feedback. Choice of Task The specific tasks used in research designs for the investigation of transitivity generally differ in five major ways. 1. Some authors (Braine, 1959; Bryant & Trabasso, 1971) have sought to control for false-positive diagnostic errors by varying the ways in which the stimuli in stick-length comparisons are presented to the subject.1 These stimulus variations include adjustments in the lengths of the sticks used, in the number of stimulus sticks presented, and in the distance between stimulus and/or test sticks and the use of a Miiller-Lyer Illusion effect to help eliminate purely perceptual judgments. 2. Riley (1975, 1976) has extended some of Trabasso's innovations to other comparative dimensions, such as happiness and niceness. 3. Another group of researchers (Harris & Bassett, 1975; Murray & Youniss, 19,68; Youniss & Furth, 1973; Youniss & Murray, 1970) have increased the range of stimuli and test comparisons to include relations, such as equal to, as well as the more common greater than.

4. Although most of the literature on transitive inference tasks is based upon nonverbal comparison, two studies (Stetson, 1974; Harris & Bassett, 1975) have tested for transitivity using a purely verbal task. In these studies children were read the premise relations in sentence form and were required to give a verbal response. 5. Task presentations have differed along the dimension of active discovery versus passive observation of the initial premise information (Bryant & Kopytynska, 1976). Below, each of these task variations are discussed in turn. One of the first attempts to control for false-positive perceptual discrimination errors was that of Braine (1959). His subjects were instructed to find candy that was consistently hidden under the longer or shorter of two upright pieces of wood. After the subjects were trained to depend on relative length as a cue to the location of the candy, they were given a series of transitivity items in which they were shown that upright A was longer than measuring stick B and that measuring stick B was longer than upright C. Then they were asked to find the candy. Only the AC comparison was tested, and according to Braine's interpretation, the age at which 507' °f tne children have transitivity is between 4 years 2 months and 5 years 5 months. Braine tried to eliminate noninferential perceptual judgments by making the difference in length between A and C very small, by placing them far apart so they could not be included simultaneously in the central field of vision, and by placing arrows on the uprights for the purpose of inducing a weak Miiller-Lyer Illusion. The illusion technique has been utilized in many studies to control for guessing and perceptual choices (Braine, 1959; McManis, 1969; Roodin & Gruen, 1970; Smedslund, 1963). 1 The main theoretical focus of this work has been to control for false-negative diagnostic errors. That is, memory training makes possible a measure that seems to be sensitive to the younger children, and this approach replaces procedures that tend to produce negative diagnoses. The control for falsepositive error referred to in the text results from methodological features of this general approach.

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Smedslund has argued that the visual illusion created by arrows making C appear to be longer than A will differentiate subjects who respond solely on the basis of perceptual cues. Presumably, these subjects should consistently choose C as the longer of the test stimuli, whereas subjects who have transitivity should be able to ignore the illusion and give the correct response, A. This procedure has been criticized by Lutkus and Trabasso (1974). They argued that use of the Muller-Lyer Illusion puts subjects into conflict. The children might actually have the ability to succeed at a transitive inference task but be so dominated by the illusion as to impair the utilization of that ability. In response to this criticism, the Piagetians would argue that if a subject cannot overcome the perceptual domination, then transitivity is not sufficiently developed to be labeled an organized structure. They would admit the possibility of false-negative diagnostic error but argue that the existence of just such an error indicates the absence of fully developed and generalizable transitive inference. In another series of experiments (Bryant, 1973, 1974; Bryant & Trabasso, 1971; DeBoysson-Bardies & O'Regan, 1973; Harris & Bassett, 197S; Lutkus & Trabasso, 1974; Riley & Trabasso, 1974; Trabasso, Riley, & Wilson, 1975) a different kind of experimental task presentation was utilized in order to prevent the transfer of absolute responses or verbal labels leading to a nontransitive solution and resulting in a false-positive diagnosis. To control for the mere parroting of the verbal labels longer for A and shorter for C, Bryant and Trabasso and their colleagues introduced more stimuli and thus more direct comparisons. The initial four direct comparisons involved five rods of different lengths and colors, with A > B, B > C, C > D, D > E. The rods were always presented in pairs and in such a way that they all appeared 1 in. (.025 m) from the top of a container. This was accomplished by boring holes of varying lengths in the wooden box. The child was thus forced to use the color differences and not the perceived lengths in making a choice. Each child was trained on the initial comparisons and tested a number

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of times on every 1 of 10 possible pairs of colored rods. The critical comparison was BD, in which both rods had been "larger" and "smaller" in some of the initial comparisons. Bryant and Trabasso (1971) hypothesized that the probability of making a correct inference on the BD test is the probability of jointly recalling the information for each of the initial training pairs, B > C and C > D. The data were consistent with this hypothesis; a further discussion of the training procedures and results are presented at a later point in this article. In summary, the five-rod array box was intended to prevent children from making direct visual comparisons and to avoid the possibility of a labeling strategy. This task seems to be a promising model for the control of falsepositive diagnostic errors due to perceptual discrimination and/or nontransitive solutions. This type of task presentation has been extended by Riley (1975, 1976) in a novel study that examined children's abilities to map four different comparative dimensions onto linearly ordered referents. Kindergarten and third-grade children were shown a row of six faces on individual cards and were instructed that these faces represented a group of children differing in height, weight, happiness, and niceness. Order was indicated by describing the relations between adjacent faces (e.g., "Mike is taller than Steve"). Each child was tested after training on all possible comparative relationships. Riley's (1975, 1976) task was designed to test the hypothesis that individuals tend to integrate premise relations such as A > B, B > C, C > D into a single linear ordering, ABCD. This task does examine the generalizability of the linear-order representation to situations with fully ordered premise information, but it does not really examine the processing used when an individual is asked to make a transitive inference from separate premise relations. In a second experiment, Riley (1975, 1976) therefore designed a vertical display in which the premise relationships were always directly observable but spatially separated. The child thus had to isolate and coordinate the premise relations. Riley hypothesized that this display would impede the construction of a single linear

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ordering and force the child to respond deductively to inferential questions. The comparative heavier was always used for the array of faces, and cards were displayed in three conditions. In the linear display condition, the cards were ordered 1, 2, 3, 4, 5, 6. In the vertical display condition, the five premise pairs were presented in a pair-wise ascending or descending vertical order, such as (1, 2), ( 2 , 3 ) , ( 3 , 4 ) , (4, S), (5, 6). In this way the resulting display formed two vertical columns running in ascending or descending order from card 1 to card S, and card 2 to card 6. In the random display condition, the five premise pairs were presented in a random vertical arrangement, such as (S, 6), (2, 3), (1, 2), (3, 4), (4, 5). The random condition was designed to prevent a child from utilizing any ordered spatial stimulus arrangement to facilitate the construction of a cognitive spatial representation. After presentation of the premise relationships, all children were tested under both a display test, in which they were instructed to find the answer while looking at the array of faces, and a memory test, in which no external display cues were provided. The hypothesis that the random condition would interfere with the use of a linear-order problem-solving strategy, was partially supported by the decision-time data in Riley's (1976) work. Another area of contention in this field of research has to do with the breadth of comparisons in both the training and testing phases of the task. Youniss and Furth (1973) observed that Bryant and Trabasso (1971) had not varied the comparison pairs in all possible ways. For example, instead of utilizing only the A > B, B > C, C> D, D > E comparisons, they might have also included such combinations as A > B > C = D > E , in which a relation of equality between two terms is embedded in a transitive chain. The criteria for diagnosis of true transitive inference (Flavell, 1977; Smedslund, 1969) include generalization of the concept in new situations; Youniss and Furth (1973) claimed that Bryant and Trabasso, by using restricted comparison relations, did not satisfy this criterion.

Youniss and Murray (1970) and Murray and Youniss (1968) varied the test comparisons in order to address the issue of generalization. In one study (Murray & Youniss, 1968) involving transitive length relations among three sticks, A, B, and C, the AC relation was tested after each of three training conditions: A > B > C , A>B = C, and A = B > C. Their findings indicate that inferential behavior was manifested by few children younger than 6 years and by 50% of the children age 7 years 6 months to 8 years 1 month. The A = B > C condition yielded the clearest age trend, whereas the A > B > C condition failed to differentiate children at any age level. Murray and Youniss (1968) suggest that "perhaps the A > B > C relation is not sufficiently sensitive to noninferential size judgments" (p. 1267). Their work seems to imply that unless the possibility of false-positives is controlled through an expansion of the comparison relationships, the conclusion that young children have transitivity is unwarranted. In contrast with these findings, Brainerd (1973b) found transitivity emerging at an earlier age using the A = B ^ C paradigm in a study of the order of acquisition of transitivity, class inclusion, and conservation. Transitivity of both length and weight was assessed in 60 kindergarten, 60 firstgrade, and 60 second-grade children. It was found that 52% of the kindergarten sample (mean age 5 years 4 months) possessed transitivity. The criterion for determining the presence or absence of each of the conceptual skills tested was six of six correct judgments. These results support the argument that the median age for the emergence of transitivity is around S to 6 years (Braine, 1959, 1964) and fail to support the Genevan position placing the median age later—at age 7 to 8 years. Further contradictory results utilizing a wider range of comparison relationships were also obtained by Harris and Bassett (197S). These authors designed a four-stick task, in which the child was presented with the premise relations A = B, B > C, C = D. The sticks were first presented visually in a modified version of the Bryant and Trabasso (1971) wooden container. An 8-in. (.2-m)

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black stick and a 4-in. (.1-m) white stick were full}' visible to the subject, while an 8-in. (.2 m) red stick and a 4-in. (.1 m) green stick were screened so that only 2 in. (.05 m) of each were visible. The child was told, for example, that the red stick, A, was the same size as the black stick, B, and that the green stick, D, was the same size as the white stick, C. The B > C premise was never verbalized but was always visible to the child. The child was asked, "Are the red and green sticks the same size?" All 20 children (mean age 4 years 4 months) made the correct assertions on all four test trials. Since two of the four sticks were fully visible to the children throughout the task presentation, and the other two sticks were partially hidden, it was possible that the subjects formed a mental image of the hidden sticks based on their stated similarity to the visible sticks. This possibility was explored by Harris and Bassett in a purely verbal task in which, again, the premises were A = B, B > C, C = D. Thirty younger children (mean age 4 years 8 months) and 29 older children (mean age S years 8 months) were given the following type of problem: "Peter is the same size as David. David is bigger than you. You are the same size as John." The test questions were "Are Peter and John the same size?" "Who is bigger, Peter or John?" A significant proportion of both the younger and older children correctly answered the final AD question. In contrast with Murray and Youniss (1968) and Youniss and Murray (1970), the studies of Harris and Bassett (1975) and Brainerd (1973) provide support for the generalizability of the Bryant and Trabasso (1971) findings to a wider range of comparative relationships. The utilization of a purely verbal task in the Harris and Bassett (1975) work is also a unique contribution with regard to task presentation in a relatively simple situation. Stetson (1974) attempted to examine the Bryant and Trabasso (1971) hypothesis with a more complex verbal task. Ninety-six children from Grades 2, 4, and 6 were initially trained to remember the six major premises of two written passages. Subjects read the paragraphs, and their retention was tested on a true-false recognition test. Stet-

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son found that the second graders took longer to reach a criterion of two consecutive correct recognitions in the training phase; yet once criterion was reached, there was no difference in performance across grade levels on the tests of inference. The final distinction in this category is between active and passive inference problems. Bryant and Trabasso's (1971) task (Lutkus & Trabasso, 1974; Riley & Trabasso, 1974; Trabasso, 1975; Trabasso, Riley, & Wilson, 1975) is a passive task in the sense that the child is simply told that A > B, B > C, and so forth. The Piagetian tower task is an active one, in that the children have to discover the premises for themselves. Bryant and Kopytynska (1976) found support for Piaget's conclusions when they too employed an active task. They argued that young children do not spontaneously use the measuring sticks in Piaget's active tower task because they have no reason to suspect that a direct comparison of the two towers by eye might be unreliable. Accordingly, they designed a measuring task in which the two quantities to be compared were invisible (i.e., holes drilled in wooden boxes). One rod, marked off in different colors, was provided as the only measuring instrument. In support of Piaget's findings, all children of ages 5 and 6 failed a version of the original tower task; however, many were successful on the hole-measuring task. Bryant and Kopytynska (1976) concluded that there is some evidence that young children can spontaneously use intervening measures and that this implies the use of inferential reasoning. Bryant (1973) also reported a previously unpublished replication of Piaget's task with five-year-olds. He found that "all children had previously done very well in a passive inference task, yet they were plainly at a loss in our active test" (p. 423). It thus appears that younger children have difficulty making a correct transitive inference response when they are required to seek out the initial measuring premises themselves. The Piagetian conception of learning and development as a process of active discovery rather than of passive training is not contradicted by these findings. However, younger children's success in passive tasks seems note-

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worthy and deserving of more attention than the Piagetians have been inclined to give it. In conclusion, the type of task chosen to investigate transitivity seems to partially differentiate cases in which successful inference has been found in younger (ages 4 and S) as opposed to older (ages 7 and 8) children. It appears that younger children can solve transitivity problems in verbal and nonverbal tasks whose features are designed to control for the possibility of false-positive error by varying stick length, distance between stimuli, and the number of stimuli tested. Younger children seem to have more difficulty in successfully completing some tasks that include relations of equality between some stimulus terms or that require an active discovery of the initial premise relations. Response Required Investigators diverge again with respect to the type of response from which transitive inference ability, or the lack of it, is to be inferred. The major controversy is over whether to rely on the child's judgment by itself (Braine, 1959; Bryant, 1973, 1974; Bryant & Kopytynska, 1976; Bryant & Trabasso, 1971; Lutkus & Trabasso, 1974; Murray & Youniss, 1968; Riley, 1975, 1976; Riley & Trabasso, 1974; Trabasso, 1975; Youniss & Murray, 1970) or to require both the judgment and the child's explanation of his or her judgment (McManis, 1969; Smedslund, 1960, 1963). Investigators who require a satisfactory explanation in addition to the judgment clearly apply a more stringent criterion; it might be expected that this diagnostic conservatism would elevate estimates of the age of emergence of transitivity. The most lively exchange in this area has been between Braine and Smedslund. It was previously noted that Braine (1959) used a nonverbal task in which subjects were instructed to find a piece of candy hidden under either the longer or the shorter of two pieces of wood. This task and Braine's conclusion that transitivity and ordinality were acquired between ages 5 and 6 were criticized by Smedslund (1963) for failure to control

for false-positive error due to nontransitive solutions. Smedslund presented replication data of his own in support of Piaget's norms; Braine (1964) replied that Smedslund's criticisms were farfetched, since there was no evidence for the occurrence of nontransitive solutions in the original task. Braine (1964) presented new data, which failed to reveal nontransitive solutions and replicated his original findings. Smedslund (1965) responded, adding the charge of "pseudomeasurement" to his earlier criticisms. Pseudomeasurement in this context refers to cases in which only one of two sticks to be compared constitutes the basis for the child's judgment. Smedslund's position was that Braine's data did not include any control for the possibility of nontransitive reasoning of the form A > B, therefore A > C. Throughout this exchange, Braine and Smedslund employed different response criteria for determining the presence of transitivity; Braine's data consisted of judgments only, whereas Smedslund required adequate explanations as well as judgments. Brainerd (1973a) argued that the discrepancy between the respective conclusions of Braine and Smedslund might be entirely attributable to this difference and that Smedslund's criterion was probably too stringent. He made the following statement: The basic proof for the presence of a given cognitive structure must, by definition, be a proof that an act of understanding appropriate to that level of structuration has taken place. By implication, if we wish to avoid being unduly conservative, then we require the minimum necessary evidence that an act of understanding has occurred. (Brainerd, 1973a, p. 177)

Explanations provide information about the nature of the underlying structures, but they may confound the diagnosis of the presence of a structure when utilized as part of the criterion measure. For example, a child may be able to make the correct judgment on a transitive inference task but be unable to verbalize an adequate explanation for the choice. The requirement of a verbal explanation as a necessary condition for diagnosis of correct transitive reasoning would exclude such children and might thus result in a false-negative error. This major difference in

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the response required of the child seems to account for some of the divergence among investigators regarding the age of emergence of transitivity. It seems possible that younger children may, in fact, have the transitivity principle and utilize it successfully but lack the verbal skills necessary to offer appropriate explanations for their responses. Smedslund (1963), Flavell and Wohlwill (1969), and others would argue conservatively that verbal explanations are necessary indicants of the presence of a true cognitive structure and that unless this criterion is employed, we may have difficulty knowing whether or not a subject used a nontransitive solution. Roodin and Gruen (1970) reported the only research that specifically investigated the judgment versus judgment-plus-explanation difference. Groups of kindergarten, firstgrade, and second-grade children were given a task similar to that of Smedslund (1963), and the Muller-Lyer Illusion was employed to correct for the possibility of perceptiondominated solutions. Half of the children at each grade level were allowed to use a memory aid for comparisons of A > B and B > C, whereas the other half were not. Almost all subjects who could verbally explain their judgments also made correct judgments, although the converse was not true. Significant age effects were found for the more stringent explanation criterion, whether or not a memory aid was used. Performance on judgments alone showed an age effect only when no memory aid was used. Subjects provided a significantly greater number of correct judgments than correct judgments plus correct explanations. Training The principal hypothesis advanced by Bryant and Trabasso and their colleagues is that children's difficulty in making transitive inferences may be more a matter of memory limitations than of logical competence level (Bryant, 1973, 1974; Bryant & Trabasso, 1971; Riley & Trabasso, 1974; Trabasso, 1975; Trabasso, Riley, & Wilson, 1975). Their standard paradigm for controlling the forgetting of initial premises and thus reduc-

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ing false-negative errors, is as follows: (a) The subject is trained on one initial pair comparison at a time in a choice discrimination task when comparative questions are asked; (b) when one pair is mastered (to a criterion of 8 out of 10 successful choices), the next one is learned; (c) when all four pairs are trained to criterion, the order of presentation is switched to a blocked-random sequence until a second training criterion of six successive correct responses on each pair is attained; and (d) testing of both adjacent and nonadjacent pairs in the transitive chain follows immediately. The child thus receives extensive memory training on the premise pairs and is tested without a long retention interval. In all studies in which this procedure was used to ensure retention of the premises, it was reliably found that children 2 or more years younger than Piaget's subjects successfully made transitive inferences, even on the critical BD comparison. This has even been found for retarded subjects whose mental age (MA) was 5 to 7 years (Lutkus & Trabasso, 1974). Compared with intellectually average subjects of similar MA level, retardates took longer to learn the initial comparisons in the training phase, but their performance on transitivity tests was above chance, with the majority of subjects perfect, and their overall performance was only slightly below the intellectually average control subjects. Bryant and Trabasso have consistently found that memory for training comparisons is highly correlated with performance on transitivity (Bryant, 1973; Bryant & Trabasso, 1971; Lutkus & Trabasso, 1974; Riley & Trabasso, 1974; Trabasso, 1975). This is especially true when the premises relevant to the BD test comparison (B > C, C > D) are examined. Although Smedslund (1963) did try to control for memory in his work by asking the subject to restate the premises immediately before the test question, Bryant and Trabasso argued that adequate initial training to ensure retention was lacking in Smedslund's study. DeBoysson-Bardies and O'Regan (1973) replicated the work of Bryant and Trabasso (1971) but varied one feature of the training phase. Children were only required to reach

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a criterion of two rather than six correct responses in the randomized block presentation of the stimulus pairs. The percentages of correct choices obtained on the nonadjacent (inference) pairs were comparable to those of Bryant and Trabasso, but the decreased learning criterion apparently affected performance on retention of the adjacent (training) pairs. DeBoysson-Bardies and O'Regan (1973) pointed out that, according to the Bryant and Trabasso (1971) hypothesis, reducing the learning criterion should also have affected the nonadjacent test pairs. In a second experiment, they further modified the training phase by always presenting the adjacent pairs in random, rather than ascending or descending, order. For example, instead of training subjects on AB, BC, CD, DE, they were trained on BC, DE, AB, CD, or AB, CD, DE, BC, and so forth. Children's performance did not suffer in this disordered training condition. However, in contrast with the children, adults performed considerably worse in the disordered condition. These results have implications for the Piaget-Trabasso debate over whether mastery or transitivity during development involves qualitative changes in ability or simply quantitative changes in information-processing skills. Further discussion of the findings of DeBoysson-Bardies and O'Regan (1973) is presented in relation to their labeling hypothesis at a later point in this review. Riley and Trabasso (1974) sought to control for two more possible sources of error in the training phase of the Bryant and Trabasso transitivity task. Although size and position cues were controlled in the original work of Bryant and Trabasso (1971), the sticks were arranged in a display box so that location relative to the ends of the array and distance between members of pairs within the array were confounded with the order of the sticks in the transitive chain. Riley and Trabasso removed these correlated cues by presenting the sticks next to each other for all pairs in both training and testing. They also varied the comparative questions used in training by asking either "Which is shorter, A or B?", or, "Which is longer, A or B?", or both, in three different experiments. In training, when both comparatives were used, sub-

jects (age 4 to S years) learned adjacent pairs faster and more often reached criterion than when only one comparative term was used. In testing, the children were more often successful when double comparative relations were used within pairs during training. Riley (1976) complemented this finding in a study of three types of comparative dimensions. She varied the comparative questions in both the training and testing phases by asking either the comparative (e.g., "Which girl is taller?") or the negative equative (e.g., "Which girl is not as tall as the other?") form of the same question. It was found that decision times to negative equative questions were slower than decision times to comparatives, but no difference in error rates to the two types of questions was observed. These findings support Piaget's claim that young children reduce the comparative to a label, such as A is long, B is not long (Piaget et al., 1960). As one of Riley's subjects reported: "You have two sizes of sticks, long and short, and you keep changing which ones are which" (Riley & Trabasso, 1974, p. 197). Riley and Trabasso went on to propose that the children studied did not use operational transitivity (coordination of the end terms via a middle term) to solve the problem. Instead, they believed that subjects integrate the initial information into an ordered, spatial array. This array is constructed during training, stored in memory, and internally scanned when inferential questions are posed. The subject thus makes an inference by observing the relation between the members of a pair in the spatial memory array and not by operational transitivity. This hypothesis begins to address the nature of the cognitive processes that actually constitute this ability we label transitive inference. In addition to extending the comparative dimensions of the Bryant and Trabasso (1971) task, Riley (1975, 1976) introduced a unique memory-aid training condition in her first experiment. In the memory-aid condition, children were presented with copies of the stimulus pictures and were told that they could use these pictures during the training phase to help them remember the

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relationships. The memory-aid pictures contained no specific information about the four comparative dimensions and thus only facilitated training if the child organized them in correspondence with the given premise relations. Riley believed that the memory aid would free the child from having to both organize the premises into a linear spatial order and memorize the sequential arrangement. Instead, the child would be able to tangibly represent the premise relations externally and could utilize short-term memory solely for the purpose of organizing the relations in the spatial array. The memory aid was always removed prior to the testing phase, thus limiting the external representation to training periods. Riley (1976) stated, Children in the Memory Aid condition made considerably fewer errors in testing than the children in the No Memory Aid condition . . . despite the facts that the children in the Memory Aid condition spent much less time in training and were not retrained, and the external mnemonic was removed during testing, (p. 9)

Feedback Part of Bryant and Trabasso's (1971) initial study included visual feedback at the end of each trial during the training phases; it was thus possible that a false-positive error could have resulted if the children solved the BD transitive question noninferentially by simply remembering that B was 6 in. (.15 m) and D, 4 in. (.1 m). The authors controlled for this possibility in their second experiment (Bryant & Trabasso, 1971) by substituting verbal for visual feedback. The full lengths of the rods were never displayed completely until the whole experiment was finished. During the experiment, only 1 in. (.025 m) of each rod was displayed. Children learned the stick-length comparisons by remembering the following verbal feedback given by the experimenter: "Yes, the red stick is longer (shorter) than the blue stick," for a correct response and "No, the red stick is longer (shorter) than the blue stick," for an incorrect response. In general, young children seem to show poorer performance with verbal feedback than with visual feedback (Trabasso, 1975); still, their

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performance is above chance level and is qualitatively comparable to that of their older peers. The effects of feedback are seldom apparent in the Piagetian research tradition, because a training phase is rarely part of the procedure. Evocability and Utilizability The practical assessment of cognitive abilities, as the foregoing discussion implies, is not a simple matter. One may gain the impression that diagnostic decision making is made virtually impossible by the many potential sources of Type I and Type II errors and the need to control for them. Smedslund (1969) has concluded, "The relationship between any set of behavioral indices and a mental process, therefore, is an uncertain one, and a diagnosis will always have the status of a working hypothesis" (p. 247). However, working hypotheses about a mental process that is assumed to be present or absent may be too simple. One way to view the practical-assessment dilemma posed by the literature on transitive inference is in terms of FlavelPs (1971, 1977) distinction between evocability and utilizability. When a child initially acquires some transitive inference ability, it may remain difficult to evoke and difficult to utilize in many situations. The research from the Piagetian group, which has identified age 8 as the age of emergence for the operation of transitivity, may actually have identified the age at which the principle can be actively and spontaneously applied in most situations. At this age, the child's levels of evocation and utilization are strong and consistent. On the other hand, Piagetian methods of diagnosis exclude those younger children for whom transitivity is evocable, given appropriate environmental conditions, but not yet spontaneously utilized in the absence of special conditions. It is reasonable to assume that newly developing cognitive skills need some environmental or practical support in order to exhibit the levels of evocability and utilization we see in the 8-year-old. Transitive inference may become available to the young child by age 4 or 5 under optimal conditions of training, feedback, and so forth; however,

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there is some reason to judge that the principle is not fully developed until age 8 or 9, when it can function reliably with minimal environmental support and across many varied situations. Conceptualization Just as the levels of utilization and evocation may undergo changes during development, it is possible and even likely that the actual processes responsible for transitive inference undergo changes, too. A major goal of current work in this area is to identify the content and sequencing of the mental processes involved in the transitive inference task (Flavell, 1977). Flavell explicitly asked the following: What actually happens, in cognitive-process terms, between problem presentation and the subject's response? When confronted with the problem, he or she presumably assembles and executes cognitive processes of some sort, processes that are integrated and sequenced in some fashion. What are those processes and how are they organized . . .? Also, why is some one particular structured set of processes assembled, rather than some other? That is, what abilities, limitations, biases, task representations, etc., within the child lead him to generate that particular set?" (pp. 227-228)

At present, we know very little of the processes that underlie transitivity. What we do know supports the linear spatial ordering hypothesis proposed by Riley and Trabasso (1974) and pursued by Trabasso and his co-workers (Trabasso, 1975). This hypothesis proposes that in the five-stick Bryant and Trabasso (1971) task, subjects gradually construct a spatial representation of the entire array in an ordered sequence, A > B > C > D > E (Bryant, 1973, 1974; Trabasso, 1975; Trabasso, Riley, & Wilson, 1975). These authors have found that children as young as 4 years seem to utilize an "ends inward" strategy of coding the array. The child first isolates the end-anchor members, sticks A and E; in this way, he/she identifies those members that can only be labeled longer (A) and shorter (E) in comparisons where they appear. Once the ends are isolated, the child begins to learn the ordered pairs in the following way: First, pairs AB and DE are encoded; this enters

the terms B and D, giving the partial ordering A > B > D > E. Then, pairs BC and CD are encoded, thus entering the final member, C. Once such an ordered representation of the whole transitive chain is stored in memory, the task of responding to test comparisons in the Bryant and Trabasso paradigm becomes almost perceptual; the tested terms can be referred to the stored array and the relation between them found, without going through a series of inferential steps. Support for this view of the cognitive basis of transitive inference was obtained in reaction time data collected by Trabasso, Riley, and Wilson (1975). It was found that reaction time for comparison of pairs decreased as a function of the distance between the members of the pair. This effect was obtained with children ages 6 and 9 and with college students. Furthermore, the reaction times for adjacent-pair comparisons were greatest for the CD pair, which is consistent with the idea that subjects have been constructing linear orderings in an ends-inward manner on the five sticks during training. These results seem to favor the linear-ordering hypothesis, at the expense of any alternative step-by-step process of logical deduction from premises to conclusion, since (a) the step-by-step class of hypotheses predicts that reaction time for comparison of pairs will increase as a function of the number of intermediate premises to be coordinated and (b) the step-by-step alternative does not explain the longer reaction time for the "middle" premise pairs. Trabasso, Riley, and Wilson (1975) hypothesized that pairs (1, 2) and (4, 5) might have been learned more easily because stick 1 (A) was labeled longer and stick 5 (E) was labeled shorter. The serial position effect found in the five-stick task then would reflect faster responses to test pairs that benefitted from this consistent labeling. To further test the linear-ordering hypothesis, Trabasso, Riley, and Wilson (1975) expanded the number of original premise sticks from five to six, thus allowing a new twostep inference comparison, the pair (2, 5). It was found that reaction time decreased and the proportion of correct responses increased as the number of "middle" terms between

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sticks increased. These results further support the linear-ordering hypothesis. Regarding the development of the process of linear ordering, Flavell (1977) noted: It is harder to achieve this sort of quasispatial internal representation in preschool children than in older subjects, e.g., more presentations of the adjacent pairs are required. Once achieved, however, preschool children can solve transitive inference problems and they appear to solve them in this very same essentially noninferential fashion, (p. 228)

This statement accepts the hypothesis of a linear-ordering process and shifts the developmental emphasis to acquisition. However, the use of linear ordering in transitive inference tasks seems to be called into question by the data of DeBoyssonBardies and O'Regan (1973). Recall that in this study, a relaxation of training criteria and/or the randomized ordering of training pairs led to a decrease in the percentage of correct responses on training comparisons but did not affect inferential comparisons. The authors proposed that children may in fact be using a labeling strategy rather than an ordered spatial array. The labeling hypothesis has three parts: First, the child learns to select the larger and smaller stick from each isolated pair during the training phase. Second, the child labels sticks within each pair as big or small. Sticks B, C, and D are given two labels each during training and thus retain no useful label, while stick A is consistently labeled big and stick E, small. Third, if a stick has been paired with another consistently labeled stick, it may acquire that label by association. By virtue of this association, two adjacent sticks may be labeled in the same way, and confusion may result from the similarity. This would explain the finding that a reduction in training criterion significantly affected performance on adjacent, rather than on nonadjacent, pairs. DeBoysson-Bardies and O'Regan (1973) attempted to test their labeling hypothesis in another experiment in which children were trained on pairs AB, BD, and DE, but stick C was left out and thus should not have acquired a useful label. Children typically did insert C between A and E when tested, but this result seems equally consis-

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tent with both the labeling and the ordering hypotheses. The authors' final test of the labeling hypothesis utilized only two separate pairs of sticks, AB and CD. Children were trained to remember that A > B and C> D and were asked to infer the relation between A and D and between B and C. The authors argued that the labeling hypothesis would predict that the child would label A and C as big and B and D as small, thus concluding that A > D and C > B. A child using an ordering strategy would, according to their analysis, conclude either that A > B > C > D, from which A > D and B > C, or that C> D > A > B, from which D > A and C > B. That is, either of the four-term spatial orderings would lead to a different prediction than the one given by the labeling hypothesis. The results supported the labeling hypothesis: Children responded that A > D and C> B. Although these findings appear to provide support for the labeling hypothesis at the expense of the ordering hypothesis, one may question the initial assumption of DeBoysson-Bardies and O'Regan that these experiments involving incomplete transitive chains of terms constitute fair tests of the two hypotheses. A labeling strategy seems plausible as an interpretation of these tasks, but the absence of transitivity in the task presentation itself might reduce the likelihood that the subject would use an ordering strategy. The DeBoysson-Bardies and O'Regan (1973) experiments may therefore have been biased in favor of an alternative strategy, such as labeling, from the outset. Harris and Bassett (197S) provided evidence that failed to support the labeling hypothesis. Their task consisted of the premises A = B, B > C, C = D, and a test of the AD relation. According to the labeling hypothesis, both A and D should be labeled same, so that children using a labeling strategy should incorrectly conclude that A = D. Children (mean age 4 years 4 months) were presented with both a visual and a verbal version of the task (a more detailed task description is given above in the Choice of Task section), and it was found that a significant number of children made the correct transitive inference, A > D.

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Riley (1975, 1976) offered the most recent and substantial support for the generalizability of the linear-ordering hypothesis. The total number of errors and the trial block of last error during training were computed for each training pair under all task conditions. A bowed serial position curve was found for the no-memory-aid condition, but the curve was almost flat for the memory-aid condition. It was also determined that all children in the "taller" and "heavier" comparison conditions and a majority of the children in the "happier" and "nicer" comparison conditions organized the pictures into a linear order. The children who did not use a single linear ordering over all the pictures chose to divide the pictures into categories and then constructed a linear order within each separate category. In her analysis of decision times during the testing phase, Riley (1975, 1976) found the same pattern as that observed by Trabasso (1975). The pairs showed an inverse relationship between distance and decision times, with the end-anchor pairs eliciting the fastest judgments. It is of interest to note that there was a tendency for the happiness and niceness conditions to be harder to learn (e.g., there were more errors and slower decisions). Riley believes that these results may suggest a developmental effect whereby children initially learn to use linear orders to represent relationships along physical or spatial dimensions before using them for relationships without such concrete properties. The tendency for decision times to be fastest for end-anchor test pairs and slower for internal pairs was found again in Riley's (1975, 1976) second experiment (described earlier), in which several conditions differing in the ease with which a linear order could be constructed and used were compared. The data from the display test revealed significantly faster decision times to end-anchor pairs in both the linear and vertical conditions but not in the random condition. However, the pattern of decision times in the random condition suggests that end anchors are isolated as a first step in the construction of a linear ordering. The analysis of internal pairs showed decreasing decision times with increasing inferential steps,

as in earlier studies. In the memory-test condition no significant end-anchor effects were found for the vertical or random conditions, but the data pattern conformed to that of the display test (i.e., end-anchor pairs were faster, and responses were more accurate than for internal pairs) in all three conditions. The decision time data for the internal pairs showed a significant distance effect for both the linear and vertical conditions but not for the random condition; once again, however, the pattern was similar to that found in the display-test condition. Riley's (1975, 197.6) experiments provide support for the idea that children prefer to represent a set of premises in an ordered sequence of terms, even when the task presentation (random condition) is intended to force the use of deductive reasoning. (Note that although we speak of deductive reasoning as a theoretical alternative to linear ordering, at this point there does not yet seem to be a process model with which to identify this alternative.) Children seemed to eliminate the difficulty of searching the display for the necessary premise information on each trial by ordering the pairs in memory. Riley suggests that the usefulness and preferability of the linear-order representation for comparative relationships reflect four major properties: (a) The representation is efficient and economical, (b) it preserves the critical comparative information so that direct reconstruction of the original premises can be accomplished and no necessary information is lost, (c) any element in the array can be easily compared with every other element or groups of elements, and (d) linear orders allow for direct comparisons. One last piece of developmental process information has been provided by Brainerd (1937b). He sought to discover the order of emergence in development of transitivity, conservation, and class inclusion of length and weight. There has been some controversy in the literature over this ordering; Brainerd found it possible to derive from the Piagetian literature two contradictory predictions regarding the order of emergence of these three abilities. The difference appears to revolve around claims concerning the development of

TRANSITIVE INFERENCE

seriation. If seriation precedes transitivity but develops synchronously with class inclusion, then class inclusion should precede transitivity, which is said to be developmentally synchronous with conservation. However, if one introduces the claim that seriation presupposes conservation, then the predicted sequence becomes conservation, then class inclusion, and then transitivity. Brainerd studied 60 white Canadian kindergarten, first-grade, and second-grade children, and 60 white American second graders in two separate investigations. All subjects were presented with a total of six tasks assessing transitivity, conservation, and class inclusion of weight and length. It was found that two sequences were evident at all three age levels included in the study: Transitivity preceded class inclusion, and conservation preceded class inclusion. Transitivity was found to precede conservation in kindergarten and first-grade subjects. Murray and Youniss (1968) and Youniss and Murray (1970) had previously found that seriation preceded transitivity. Therefore, we have a (transitively inferred) predicted acquisition sequence as follows: seriation, then transitivity, then conservation, and then class inclusion, for weight and length. Summary There seems to be strong evidence that children ranging in age from 4 to 10 years, mentally retarded adolescents, and college students use similar strategies of constructing linear orderings from paired and ordered information, of storing these representations, and of using them to make inferences about comparative relations. They appear to use an ends-inward strategy when the end sticks are easiest to isolate and recall. Trabasso (197S) hypothesized that the shorter reaction time of the older child or adult probably reflects a more strongly associated set of linearly ordered codes, and not qualitative differences in cognitive abilities. The relatively poorer performance of younger children has consistently been in those conditions in which feedback was linguistic, in which only a single comparative (e.g., longer or shorter) was used in training, and in which

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the task involved an active discovery of the basic premise relationships. Generally, when young children (age 4 to 6) have been provided with visual referents equivalent to a linguistic message, such as visual feedback, a visual display, or some other external memory aid, they reach the adult criterion. According to Trabasso (197S), Since adults seem to use the same representations and mental operations as children across a variety of tasks, once they commit the premises to memory, we are forced to conclude that the cognitive processes of children and adults are very much alike. Seriation and the use of its products seem to underlie transitivity or at least the ability to make inferences of a transitive nature. The locus of failure, we suggest, is the greater contextual dependency of language for younger children, (p. 169)

It is important that Piagetian developmental theorists not overlook the substantial process-oriented work of the Trabasso group. As Flavell (1977) contended, this type of analysis is essential if we are to explain and describe the course of cognitive growth. Future diagnostic research on the transitive inference principle will benefit by extending a process-oriented approach beyond Bryant and Trabasso's original paradigm. Situations that are purely verbal in nature or require active measurement and participation of the subject are of particular interest. Will the memory factor still remain an important component if the subject is allowed to measure and discover the comparative relations for himself or herself? It will also be interesting to determine whether subjects who have linearly ordered the Bryant and Trabasso task stimuli and have successfully responded to all pair combinations can transfer that training to new tasks of varying similarity to the basic model. It seems that persons of all ages can use a linear-order strategy to perform successfully on simple transitive inference tasks, but we do not yet know whether there are or are not any qualitative differences in the logical abilities of younger and older children in relation to transitivity. Although the Piagetian theorists have attempted to determine and validate the age of emergence for transitive reasoning, they have yet to study in detail the actual cognitive processes that comprise this skill. If

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children reason in a qualitatively different manner than adults, then what are the differences, and how can they be demonstrated and measured in transitive inference and other tasks? It now seems appropriate for workers in the field of cognitive development to adopt a modelling approach to children's reasoning. In particular, it would be useful for Piagetians to accept the challenge of formulating a theoretical model of transitive inference that would account for the available data as well as does the linear-order hypothesis and lead to new predictions. The quest for sensitive assessment has been furthered by the efforts of researchers to understand the development of transitive inference. In the work reviewed here one can discern progress not simply in the accuracy with which psychologists are able to make a diagnostic judgment but in psychology's understanding of that which is being judged. Furthermore, there are encouraging signs that workers in cognitive development recognize that the one is dependent upon the other. References Braine, M. D. S. The ontogeny of certain logical operations: Piaget's formulations examined by nonverbal methods. Psychological Monographs, 1959, 73(5, Whole No. 475). Braine, M. D. S. Development of a grasp of transitivity of length: A reply to Smedslund. Child Development, 1964, 35, 799-810. Brainerd, C. J. Judgments and explanations as criteria for the presence of cognitive structures. Psychological Bulletin, 1973, 79, 172-179. (a) Brainerd, C. J. Order of acquisition of transitivity, conservation, and class inclusion of length and weight. Developmental Psychology, 1973, 8, 105116. (b) Bryant, P. E. What the young child has to learn about logic. In R. A. Hinde & J. Stevenson-Hinde (Eds.), Constraints on learning. New York: Academic Press, 1973. Bryant, P. E. Perception and understanding in young children: An experimental approach. New York: Basic Books, 1974. Bryant, P. E., & Kopytynska, H. Spontaneous measurement by young children. Nature, 1976, 260, 773. Bryant, P. E., & Trabasso, T. Transitive inferences and memory in young children. Nature, 1971, 232, 456-458.

DeBoysson-Bardies, B., & O'Regan, K. What children do in spite of adults' hypotheses. Nature, 1973, 246, 531-534. Flavell, J. H. The developmental psychology of Jean Piaget. New York: Van Nostrand, 1963. Flavell, J. H. Stage-related properties of cognitive development. Cognitive Psychology, 1971, 2, 421453. Flavell, J. H. Cognitive Development. Englewood Cliffs, N.J.: Prentice-Hall, 1977. Flavell, J. H., & Wohlwill, J. F. Formal and functional aspects of cognitive development. In D. Elkind & J. H. Flavell (Eds.), Studies in cognitive development. New York: Oxford University Press, 1969. Harris, P. L., & Bassett, E. Transitive inferences by four-year-old children? Developmental Psychology, 1975, 11, 875-876. Langer, S. K. An introduction to symbolic logic (3rd rev. ed.) New York: Dover, 1967. Lutkus, A., & Trabasso, T. Transitive inferences by preoperational, retarded adolescents. American Journal of Mental Deficiency, 1974, 78, 599-606. McManis, D. L. Conservation and transitivity of weight and length by normals and retardates. Developmental Psychology, 1969, 1, 373-382. Murray, J. P., & Youniss, J. Achievement of inferential transitivity and its relation to serial ordering. Child Development, 1968, 39, 12591268. Piaget, J., Inhelder, B., & Szeminska, A. The child's conception of geometry. New York: Basic Books, 1960. Riley, C. A. Representation and use of comparative information and inference making by young children (Doctoral dissertation, Princeton University, 1975). Dissertation Abstracts International, 1976, 36, 4210B-4211B. (University Microfilms No. 76-00264) Riley, C. A. The representation o.f comparative relations and the transitive inference task. Journal of Experimental Child Psychology, 1976, 22, 122. Riley, C. A., & Trabasso, T. Comparative, logical structures, and encoding in a transitive inference task. Journal of Experimental Child Psychology, 1974, 17, 187-203. Roodin, M. L., & Gruen, G. E. The role of memory in making transitive judgments. Journal of Experimental Child Psychology, 1970, 10, 264-275. Smedslund, J. The problem of "what is learned." Psychological Review, 1953, 60, 157-158. Smedslund, J. Transitivity of preference patterns as seen by pre-school children. Scandinavian Journal of Psychology, 1960, 1, 49-54. Smedslund, J. Development of concrete transitivity of length in children. Child Development, 1963, 34, 389-405. Smedslund, J. The development of transitivity of length: A comment on Braine's reply. Child Development, 1965, 36, 577-580. Smedslund, J. Psychological diagnostics. Psychological Bulletin, 1969, 71, 237-248.

TRANSITIVE INFERENCE Stetson, P. C. Verbal transitivity in children (Doctoral dissertation, University of Delaware, 1974). Dissertation Abstracts International, 1974, 35, 2064A-2065A. (University Microfilms No. 74-22, 218) Trabasso, T. Representation, memory, and reasoning: How do we make transitive inferences? In A. D. Pick (Ed.), Minnesota Symposia on Child Psychology (Vol. 9). Minneapolis: University of Minnesota Press, 197S. Trabasso, T., Riley, C. A., & Wilson, E. G. The representation of linear order and spatial strate-

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gies in reasoning: A developmental study. In R. J. Falmagne (Ed.), Reasoning: Representation and process in children and adults. Hillsdale, N.J.: Erlbaum, 1975. Youniss, J., & Furth, H. G. Reasoning and Piaget. Nature, 1973, 244, 314-316. Youniss, J., & Murray, J. P. Transitive inference with nontransitive solutions controlled. Developmental Psychology, 1970, 2, 1969-1975. Received August 22, 1977