Transitive inference in rats (Rattus norvegicus).

Although Piagetian theory proposes that the ability to make transitive inferences is confined to humans above age 7, recent evidence has suggested that this logical ability may be more broad based. In nonverbal tests, transitive inference has been demonstrated in preschool children and 2 species of nonhuman primates. In these experiments, I demonstrate evidence of transitive inference in rats (Rattus norvegicus). I used an ordered series of 5 olfactory stimuli (A < B < C < D < E) from which correct inferences were made about the novel B versus D pair. Control procedures indicated that performance did not depend on the recency with which the correct answer was rewarded during training and may be disrupted by the addition of logically inconsistent premises (F > E and A > F). The possibility that logical transitivity may reflect a form of spatial paralogic rather than formal deductions from a syllogistic-verbal system is discussed.     

On the basis of early transitivity judgments

This research had two aims. The first was to test three explanations of performance on N-term series tasks by young children: the labeling model of B.DeBoysson-Bardies and K. O'Regan (1973), Nature (London), 246, 531–534, the sequential-contiguity model of L. Breslow (1981, Psychological Bulletin, 89, 325–351), and the ordered array or image model of C. A. Riley and T. Trabasso (1974, Journal of Experimental Child Psychology, 17, 187–202). In the first experiment, 5-year-old children were taught additional premises which would interfere with labeling and sequential-contiguity processes, but not with forming an ordered array. Reasoning performance was essentially comparable to previous results with the paradigm, thus supporting the ordered array model. The second aim was to reexamine children's ability to learn sets of premises which can be assembled into an ordered array, since there was reason to believe that previous studies had created false positives. In the second experiment, 3- to 7-year-old children were taught either overlapping (a > b, b > c, …) or nonoverlapping (a > b, c > d, …) premises. Overlapping premises can be integrated into an ordered array (a, b, c, d, e), but nonoverlapping premises cannot. However, the overlapping condition proved more difficult, and the success rate for preschoolers ($\text{3}\text{1}\text{2}\text{-}\text{to 4}\text{1}\text{2}\text{-year-olds}$) was of zero order. This raises doubts about their ability to learn a set of premises of the kind required for transitive inference. These doubts were strengthened by the third experiment which showed that when premises were not presented in serial order, preschool ($\text{3}\text{1}\text{2}\text{-}\text{to 4}\text{1}\text{2}\text{-year-old}$) children could not learn the premises of an N-term series task.     

4—6岁幼儿空间方位传递性推理能力的发展

该研究旨在探查4、5、6岁幼儿空间上下和前后方位传递性推理能力的发展水平及不同实验条件(一致、模糊、冲突)下儿童的传递性推理能力。为了尽量降低记忆对推理的影响,要求幼儿在前提呈现的条件下按前提方位关系进行传递性推理操作。被试为幼儿园4、5、6岁组儿童各24人,其中男女各半。主要结果表明：4岁幼儿开始萌发空间前后和上下方位的传递性推理能力;从4岁到6岁,“上下”方位传递性推理能力的发展优于“前后”方位;4—6岁幼儿还不能完全摆脱知觉干扰因素的影响,形成稳定的传递性推理能力。     

Getting one step closer to deduction: Introducing an alternative paradigm for transitive inference

Transitive inference is claimed to be “deductive”. Yet every group/species ever reported apparently uses it. We asked 58 adults to solve five-term transitive tasks, requiring neither training nor premise learning. A computer-based procedure ensured all premises were continually visible. Response accuracy and RT (non-discriminative nRT) were measured as is typically done. We also measured RT confined to correct responses (cRT). Overall, very few typical transitive phenomena emerged. The symbolic distance effect never extended to premise recall and was not at all evident for nRT; suggesting the use of non-deductive end-anchor strategies. For overall performance, and particularly the critical B?D inference, our findings indicate that deductive transitive inference is far more intellectually challenging than previously thought. Contrasts of our present findings against previous findings suggest at least two distinct transitive inference modes, with most research and most computational models to date targeting an associative mode rather than their desired deductive mode. This conclusion fits well with the growing number of theories embracing a “dual process” conception of reasoning. Finally, our differing findings for nRT versus cRT suggest that researchers should give closer consideration to matching the RT measure they use to the particular conception of transitive inference they pre-held.     

3～5岁幼儿在视野阻隔任务中的长度传递性推理

设计视野阻隔任务,控制了短时记忆和直接比较的两难问题,考察3～5岁幼儿长度传递性推理能力的发展趋势和传递性推理这一认知过程的发展序列。以2(间隔长度)×2(摆放方式)×4(年龄)的三因素混合设计,研究了240名幼儿,结果表明:(1)3～4岁是长度传递性推理能力的快速发展期,4岁以后的幼儿大部分具备了这一能力,发展速度趋于平缓;(2)任务类型影响推理成绩;(3)基于传递关系的推理依据与推理总成绩显著正相关。研究认为:年仅4岁幼儿具备传递性推理能力蕴含教育价值;传递性推理的认知过程经历了四个可辨别的发展序列,依次主要发生在36～42个月、43～48个月、49～54个月和55～60个月的幼儿当中。     

The Development of Transitivity of Length in Young Children

This research examined whether developmental changes occur in young children's abilities to learn transitive information. Children 4 years 6 months to 5 years 6 months and 6 to 7 years were asked to learn a series, A > B > C, or a nonseries, A > B, C > D. Younger children took longer to learn the series than the nonseries. Children learning the series correctly answered the inference question, but could not distinguish between situations in which it was possible or impossible to make inferences. The results suggest that there are developmental changes in transitivity abilities and that transitivity is a concept that emerges gradually.     

5~6岁儿童传递推理能力的发展特点

通过有效的实验设计,在控制儿童记忆容量影响的前提下,考察了5~6岁儿童在长度传递推理和重量传递推理能力上的发展水平。结果表明:(1)5~6岁是儿童传递推理能力迅速发展的时期。在大概6岁左右,大多数儿童已经能够进行真正意义上的传递推理。(2)此年龄儿童在长度传递推理和重量传递推理能力上不具有显著差异。(3)条件排列的一致与不一致对于被试的推理成绩并不具有显著影响。     

Transitive and pseudo-transitive inferences

Given that A is longer than B, and that B is longer than C, even 5-year-old children can infer that A is longer than C. Theories of reasoning based on formal rules of inference invoke simple axioms ("meaning postulates") to capture such transitive inferences. An alternative theory proposes instead that reasoners construct mental models of the situation described by the premises in order to draw such inferences. An unexpected consequence of the model theory is that if adult reasoners construct simple models of typical situations, then they should infer transitive relations where, in certain cases, none exists. We report four studies corroborating the occurrence of these "pseudo-transitive" fallacies. Experiment 1 established that individuals' diagrams of certain non-transitive relations yield transitive conclusions. Experiment 2 showed that these premises also give rise to fallacious transitive inferences. Experiment 3 established that when the context suggested alternatives to the simple models, the participants made fewer errors. Experiment 4 showed that tense is an important aspect of meaning which affects whether individuals draw transitive conclusions. We discuss the implications of these results for various theories of reasoning.     

The role of cues to differential absolute size in children's transitive inferences

To investigate the role that "nonlogical" cues might play in transitive inference, 6- and 7-year-olds were given a three-term transitive task in which perceptual cues to differential absolute size were either present or absent. Relationships between the taught premises and the relational information that was physically present were manipulated using four basic conditions: "congruent," "inverse," "pretended," or "persuaded." Both age groups showed identical overall premise memory, but the younger group tended to reason more on the basis of the perceptual information rather than on the successfully encoded premise information. Contrasts between the various conditions showed that categorical effects can be circumvented in three-term problems with appropriate controls, that there may be qualitative as well as quantitative differences in transitive inference with age, and that transitive inference is not based solely on memory. The findings also indicate that, although 7-year-olds are competent in "logic-based" transitive inference, they experience great difficulty on tasks involving pretend information.     

Learning strategy differentially impacts memory connections in children and adults

Even once children can accurately remember their experiences, they nevertheless struggle to use those memories in flexible new ways—as in when drawing inferences. However, it remains an open question as to whether the developmental differences observed during both memory formation and inference itself represent a fundamental limitation on children's learning mechanisms, or rather their deployment of suboptimal strategy. Here, 7–9-year-old children (N = 154) and young adults (N = 130) first formed strong memories for initial (AB) associations and then engaged in one of three learning strategies as they viewed overlapping (BC) pairs. We found that being told to integrate—combine ABC during learning—both significantly improved children's ability to explicitly relate the indirectly associated A and C items during inference and protected the underlying pair memories from forgetting. However, this finding contrasted with implicit evidence for memory-to-memory connections: Adults and children both formed A-C links prior to any knowledge of an inference test—yet for children, such links were most apparent when they were told to simply encode BC, not integrate. Moreover, the accessibility of such implicit links differed between children and adults, with adults using them to make explicit inferences but children only doing so for well-established direct AB pairs. These results suggest that while a lack of integration strategy may explain a large share of the developmental differences in explicit inference, children and adults nevertheless differ in both the circumstances under which they connect interrelated memories and their ability to later leverage those links to inform flexible behaviours.

Research Highlights

• Children and adults view AB and BC pairs related through a shared item, B. This provides an opportunity for learners to connect A–C in memory.
• Being encouraged to integrate ABC during learning boosted performance on an explicit test of A–C connections (children and adults) and protected from forgetting (children).
• Children and adults differed in when implicit A–C connections were formed—occurring primarily when told to separately encode BC (children) versus integrate (adults), respectively.
• Adults used implicit A–C connections to facilitate explicit judgments, while children did not. Our results suggest developmental differences in the learning conditions promoting memory-to-memory connections.

     

小学儿童一维空间方位传递性推理能力的发展

研究了小学儿童一维空间方位传递性推理能力的发展水平及认知策略 ,同时 ,对心理模型理论进行了检验。被试为城市中等小学 7岁、9岁、11岁儿童各 2 4名 ,男女各半。 4种实验任务分别为三前提单模型、三前提双模型、四前提单模型和四前提双模型。采用个别实验 ,儿童在前提呈现的情况下进行推理。主要研究结果 :(1)从小学 7岁到 11岁 ,儿童的一维空间方位传递性推理能力明显提高 ,7岁儿童初步形成了一维空间方位推理能力 ,9岁和 11岁基本具有了这种能力 ;(2 )随着年龄增长 ,使用模型建构策略解决问题的儿童人次越来越多 ,绝大部分 11岁儿童都能使用这一策略进行推理。但即使儿童使用了模型建构策略 ,他们的推理成绩也没有反映出模型数量所造成的任务难度差异 ,即不符合心理模型理论关于模型数量的主要预期。     

Children''s ability to make transitive inferences: The importance of premise integration and structural complexity

Four experiments investigated 4- to 6-year-olds' transitive inferences. In Experiments 1–3, there was a nonmapping condition in which inferences were made either about stacked blocks, or about sticks ordered left to right. In the mapping condition, inferences were made by mapping either from blocks to sticks, or the reverse. In Experiments 2–4, relational complexity was manipulated by requiring either 1 or 2 premise relations to be processed in a single decision. Mapping was harder than nonmapping, but relational complexity was the main source of variance, with 2 relations being harder in both mapping and nonmapping conditions. The percentage of participants integrating 2 relations in a single decision was estimated at 20% at age 4, 53% at age 5 and 57% at age 6, suggesting gradual development of transitive inference ability. Results suggest that relational complexity has a strong effect on transitive inference in 4- to 6-year-old children.     

Sequential analysis of visual habituation in preschool children

The study breaks down the data from a preschool (mean age of students, 63 months) visual habituation experiment using a second-by-second analysis of the entire process, including fixation time and onsets for habituation and recovery trials and for each intertrial interval. The results parallel those suggested by Cohen, DeLoach, and Rissman (Child Development, 1975, 46, 611–617), for infants and are described using a two-process model of visual attention.     

TRANSITIVE INFERENCE IN RATS:

Abstract— Rats were trained to discriminate between boxes covered with distinctive odors There were six stimulus odors, labeled A through F, and the problems learned formed the five premises A+B−, B+C−, C+D−, D+E−, and E+F− Combming the premises, the relative values of the stimuli were A>B>C>D>E>F In two experiments, linear arrangement groups learned these premises with Boxes A through F placed in a linear spatial sequence Nonlinear groups had boxes either randomly changed from one position to another (Experiment 1) or placed m a circular arrangement (Experiment 2) Tests of transitive inference between the B and D stimuli were carried out in an environment different from that m which premise training took place Only the groups trained with a linear arrangement of boxes showed evidence of transitive inference These findings offer support for a spatial coding hypothesis of transitive inference in animals     

Disjunctive illusory inferences and how to eliminate them

The mental model theory of reasoning postulates that individuals construct mental models of the possibilities in which the premises of an inference hold and that these models represent what is true but not what is false. An unexpected consequence of this assumption is that certain premises should yield systematically invalid inferences. This prediction is unique among current theories of reasoning, because no alternative theory, whether based on formal rules of inference or on probabilistic considerations, predicts these illusory inferences. We report three studies of novel illusory inferences that depend on embedded disjunctions—for example, premises of this sort: A or else (B or else C). The theory distinguishes between those embedded disjunctions that should yield illusions and those that should not. In Experiment 1, we corroborated this distinction. In Experiment 2, we extended the illusory inferences to a more stringently controlled set of problems. In Experiment 3, we established a novel method for reducing illusions by calling for participants to make auxiliary inferences.     

“TRANSITIVE INFERENCE” IN MULTIPLE CONDITIONAL DISCRIMINATIONS

We used multiple conditional discriminations to study the inferential abilities of pigeons. Using a five-term stimulus series, pigeons were trained to respond differentially to four overlapping pairs of concurrently presented stimuli: A+B?, B+C?, C+D?, and D+E?, where plus and minus indicate the stimulus associated with reinforcement and extinction, respectively. Transitive inference in such situations has been defined as a preference for Stimulus B over Stimulus D in a transfer test. We measured this and other untrained preferences (A vs. C, A vs. D, B vs. E, etc.) during nonreinforced test trials. In three experiments using a novel, rapid training procedure (termed autorun), we attempted to identify the necessary and sufficient conditions for transitive inference. We used two versions of autorun: response-based, in which the subject was repeatedly presented with the least well-discriminated stimulus pair; and time-based, in which the subject was repeatedly presented with the least-experienced stimulus pair. In Experiment 1, using response-based autorun, we showed that subjects learned the four stimulus pairs faster than, but at a level comparable to, a previous study on transitive inference in pigeons (Fersen, Wynne, Delius, & Staddon, 1991), but our animals failed to show transitive inference. Experiments 2 and 3 compared time- and response-based autorun. Discrimination performance was maintained, but transitive inference was observed only on the second exposure to the response-based procedure. These results show that inferential behavior in pigeons is not a reliable concomitant of good performance on a series of overlapping discriminations. The necessary and sufficient conditions for transitive inference in pigeons remain to be fully defined.     

Are children any more logical than monkeys on the five-term series problem?

Six-year-old children were tested on several versions of the five-term transitivity problem as used by B. O. McGonigle and M. Chalmers (1977, Nature (London), 267, 694–696) with squirrel monkeys as subjects. Both binary and triadic versions of the tests were administered in both verbal and nonverbal modes to help determine whether or not any major procedural differences between the monkey version and that used conventionally in research with children might account for the monkey's apparently nonlogical solution of the problem. The main result is that children showed very similar response profiles to that of monkeys in all the conditions used. In addition, “labeling”, direct seriation, and “association” post-tests suggest that nonlogical strategies can underwrite ostensibly impeccable transitive “reasoning” in child as well as monkey.     

The link between deductive reasoning and mathematics

ABSTRACT

Recent studies have shown that deductive reasoning skills (including transitive and conditional inferences) are related to mathematical abilities. Nevertheless, so far the links between mathematical abilities and these two forms of deductive inference have not been investigated in a single study. It is also unclear whether these inference forms are related to both basic maths skills and mathematical reasoning, and whether these relationships still hold if the effects of fluid intelligence are controlled. We conducted a study with 87 adult participants. The results showed that transitive reasoning skills were related to performance on a number line task, and conditional inferences were related to arithmetic skills. Additionally, both types of deductive inference were related to mathematical reasoning skills, although transitive and conditional reasoning ability were unrelated. Our results also highlighted the important role that ordering abilities play in mathematical reasoning, extending findings regarding the role of ordering abilities in basic maths skills. These results have implications for the theories of mathematical and deductive reasoning, and they could inspire the development of novel educational interventions.     

On the emergence of the discriminative mode for transitive-inference

Transitive-inference tasks play increasing roles in many areas of cognitive science. However, discrepancies between the two competing paradigms (classical-Piagetian and IP-paradigm) imply that one paradigm fails to index its target (deductive) competence. Here, potential flaws of one were addressed using considerations from the other. Children of 5–7 years (N=216) solved 3-term transitive problems, some implying a single largest item, and some not. Findings indicate a discriminative competence (a Transitive-Switch) at least 2 years before competence in actually deducing correct solutions. Interestingly, although the transitive-switch was constant across age with both memory and inference increasing with age, it was nevertheless a much stronger predictor of transitive performance than either memory or age. A predictive relationship between memory and inference held for the 5- and 6-year-olds but not the 7-year-olds; suggesting as it approaches maturity, transitive-inference becomes independent of memory. Coincidental with this latter finding, only the 7-year-olds’ transitive performance approximated a deductive competence. As the initial validation of the IP-paradigm rested on acceptance of its age estimate of 4 years, its claim to index deduction must now be revisited.     

小学儿童在不同刺激材料中的左右空间方位传递性推理能力及策略分析

采用定性和定量相结合的研究方法,从图形刺激材料和文字刺激材料入手,探查了7岁、9岁、11岁儿童的一维空间方位传递性推理能力的发展水平及其认知策略。主要研究结果表明:(1)小学儿童的一维空间方位传递性推理能力从7岁到9岁,从9岁到11岁都有显著性提高。7岁儿童处于能力的初步形成期,9岁儿童是能力的提高和发展期,11岁儿童已基本具有了一维空间方位单模型条件下的传递性推理能力;(2)小学儿童在图形刺激材料条件下的推理成绩显著优于文字刺激材料条件下的推理成绩;(3)前提数量的增加没有对小学儿童的推理造成显著性影响;(4)随着儿童年龄的增长,模型建构策略的使用者越来越多,但在图形刺激材料和文字刺激材料下,儿童解决问题的策略有所不同。在图形刺激材料下,大部分7岁和9岁的儿童采用知觉策略,11岁儿童中只有少部分人使用这一策略。在文字刺激材料下,模型建构策略是儿童推理的主要策略。少部分7岁儿童使用模型建构策略,大部分9岁儿童和绝大部分11岁儿童使用模型建构策略来解决问题。