Transitivity, sometimes referred to as transitive inference, is the ability to understand the relational properties between objects or concepts. For instance, if a German Shepherd is a dog, and a dog is a mammal, then a German Shepherd must be a mammal.
Transitivity is a concept from Piaget’s theory of cognitive development. According to Piaget, children develop transitivity during the concrete operational stage, which occurs between the ages of 7 and 11 or 12.
Transitivity in Piaget’s Psychology
In the words of Inhelder and Piaget (1958):
“…if x implies y and if y implies z, then x implies z.”
This cognitive ability is:
“…acquired during the concrete stage for serial ordering of asymmetrical transitive relations: A < B, B < C, thus A < C” (p. 306).
Bouwmeester et al. (2007) define a transitive inference task succinctly:
“A transitive reasoning task requires the inference of an unknown relationship between two objects from the known relationships between each of these objects and a third object” (p. 42).
Successfully performing this type of task demonstrates the ability to think logically, which Piaget believed key to the development of more advanced cognitive skills.
At What Age and Developmental Stage do Children Develop Transitivity?
According to Piaget’s (1959) theory of children’s cognitive development, transitivity develops during the concrete operational stage (ages 7 to 11 or 12 years old).
In this stage, the child is no longer limited to making judgements solely based on the perceptual properties of stimuli, but can instead use logic and perform purely mental operations to make inferences.
Children also begin to overcome the limitations of egocentrism. They can see physical objects and certain issues from the point of view of others.
This is part of a foundation for the development of social skills and emotional intelligence.
- Who’s Taller: A teacher can help students understand transitive inference by choosing different students of varying heights and then stating how they are related. For example: If John is taller than Mary, and Mary is taller than Sue, then is John taller than Sue? Children that have acquired the ability to use transitive inference will answer in the affirmative.
- Who’s the Fastest: Children love to compare how fast they can run. To help children develop transitivity, the teacher can ask the students a question: If Billy is faster than Jane, and Jane is faster than everyone else in the class, then who is the fastest?
- To Buy or Not to Buy: Transitive inference also happens while shopping. For example: Brand X’s phone warranty is worse than Brand Y’s, and Brand Y’s warranty is worse than Brand Z’s. So, which brand’s phone has the best warranty?
- In Choosing Clothes: Cindy likes the color purple. Cindy hates wearing boots. Will Cindy buy a pair of purple boots?
- In the Formation of Social Circles: Friendships at all ages can be complicated and do not always adhere to linear analysis. For example, if A and B are friends, and B and C are friends, then is it logical to conclude that A and C are friends?
- When Choosing Which Race Car to Play with: Presenting a transitivity task to children that involves objects they play with frequently can make the task easier to process. So, a teacher or parent can ask a child which car they want to play with by asking: “If the blue car is slower than the green car, and the green car is slower than the red car, then which car do you want to play with?”
- A Set of 4 Relational Operations: This video demonstrates transitive inference using multiple relational statements between A, B, C, D, and E. The child is presented with various operations and then asked several questions that require logic that transfers from one relation to the next.
- In Choosing Which Party to Attend: Adults also use transitive inference when making social decisions. If Party A is far away but will be more fun than Party C, and Party B is near but will be more boring than Party C, then, assuming that fun is the most important dimension, which party is the best choice?
- In Choosing a College Major: Major X will make more money than Major Y. Major Y will make more money than Major Z. So, which major will a student choose? Well, this is the problem with transitive inference that only occurs along a singular dimension. Like most decisions in life, there are multiple factors in the decision-making equation. To make matters more complicated, each factor carries a different weight of importance.
- In Choosing a Beverage: When entering a convenience store after an intense workout, you are met with dozens of options. Each product comes in a different size, has a different price point, and contains various combinations of healthy and unhealthy ingredients. The kind of mental calculus required to make a choice like this is a kind of quantum transitivity.
Case Studies of Transitivity
1. Transitive Inference in Squirrel Monkeys
Believe it or not, transitive inference has been well-researched in animal studies, including pigeons, chimpanzees, rats, and squirrel monkeys.
The typical procedure, as depicted in the above video, involves presenting two different colored cups. One has a small piece of food under it (Cup A > Cup B).
Once the animal finds the food, another set of cups is presented. This time, the cup that has the food is the same color as the cup in the previous trial that did not have food (Cup B > Cup C).
This process is repeated several times: (Cup C > Cup D) and (Cup D > Cup E).
At the end of the trials, the animal is then presented with two colored cups: Cup B and Cup D.
Quite consistently, the monkeys would choose Cup B over Cup D.
The results are explained as being an example of transitive inference. You can hear the researcher’s explanation in the video starting at 3:52.
2. Social Relations Transitivity
Studies on transitivity usually involve serial ordering objects on a singular dimension such as length. However, children and adults exist in a social context, which is nonlinear and multi-dimensional.
Markovits and Dumas (1999) examined if children use transitivity to resolve social scenarios.
They designed friendship scenarios that involved a child wanting to invite two friends to a party.
Children were presented with images of children depicting that A is friends with B, B is friends with C, and D is friends with E.
The main question posed to the children in the study was which child would A invite (aside from the obvious choice B).
A total of 302 children in grades 1-4 in Montreal participated.
“Children’s judgments about friendship were coded as transitive if they predicted that children B and C would be invited to the party,” with results showing that “There was an overall increase in transitive responding to the friendship questions over grade level” (p. 103).
The authors concluded that:
“…children can and do make transitive judgments about friendship relations,” and “the developmental pattern of change indicates that their tendency to make these kinds of judgments increases regularly with age” (p. 105).
3. Gender and Age
Gender differences in children’s transitive reasoning have been rarely examined. This is surprising given the nature of most serial ordering tasks that involve perceptual cues of length and height, both relying on spatial reasoning.
Wright and Smailes (2015) investigated both age and gender on transitive inference by presenting 117 male and female children ages 5-8 years old with 5 different tasks.
The five tasks presented two images depicting relationships between animals, household items, cars, balls, and Finding Nemo.
For example, one card showed a sheep winning a race against a horse (A > B), while the second showed the horse winning a race against a pig (B > C).
The child was then asked, of the three animals, which was fastest.
The results showed that:
“…boys tended to perform around 7% higher than did girls” (p. 971).
Regarding age trends,
“…there was around a 10% improvement in performance between the 6- and 7-year-olds, and a further 6% improvement between ages 7 and 8 years” (p. 971).
The gender result “may be due to spatial abilities taking priority over verbal abilities between ages 6 and 8 years” (p. 974).
4. Verbally Presented Transitive Relationships
As is the case sometimes in psychological research, methodological issues can take center stage in the literature. This is the case with early research on transitivity (Odom & Coon, 1967).
For instance, some have argued that young children, ages 5 and 6 years old, do not understand relational terms such as longer than or more than (Brainer, 1964). Hence, it’s not surprising that so many studies found that children at this age did not show transitivity.
The issue of using actual physical objects also clouds interpretation of results. Using physical objects makes perceptual qualities salient.
This can be distracting, disrupt children’s thinking, and doesn’t exercise the same cognitive processes involved in logical reasoning.
Enter Odom and Coon’s study. Participants were 95 second graders from middle class families in Nashville, Tennessee.
A female experimenter presented 6 different problems verbally using a variety of common objects and comparison terms.
“The results of the present study indicate that with such distracting cues absent, children in the age range of 7 to 8 are quite capable of dealing with verbally presented transitive relationships” (p. 306).
5. Transitive Inference by Human Infants
Although Piaget initially asserted that transitive inference (TI) developed in the concrete operational stage (Inhelder & Piaget, 1958), which begins around age 7 years old, research has found simple versions of this ability in children age 4 years old (Bryant & Trabasso, 1971).
Gazes et al. (2015) outlined previous studies which have demonstrated that human infants are capable of making rudimentary inferences.
For example, infants 7 to 10 months old can distinguish between ascending and descending line lengths (e.g., Brannon, 2002; de Hevia & Spelke, 2010), and prosocial from antisocial behaviors (Hamlin & Wynn, 2011).
Therefore, there is reason to believe that infants may be capable of transitive inference.
Two groups of infants 10-13 months-old watched a video which depicted social dominance among animal puppets (bear > elephant; hippo > bear).
Infants then watched another video of the two puppets that had not interacted in the first video.
This video either depicted a dominance hierarchy consistent (hippo > elephant) or inconsistent (elephant > hippo) with the hierarchy depicted in the first video.
“Infants looked longer to incongruent than congruent dominance interactions in the experimental condition, suggesting that they were using TI to infer dominance relations” (p. 5).
Transitive inference refers to using logic to infer relationships between two objects or concepts based on shared relationship with other objects or concepts.
This is a key cognitive milestone in Piaget’s theory of cognitive development which occurs in the concrete operational stage.
Although this stage ranges from ages 7 to 11, research has demonstrated transitive inference in younger children and infants.
Other studies have demonstrated that non-human animals utilize transitive inference as well.
Transitive inference is exercised in everyday decision-making, while a more complex multi-faceted form of analysis takes place in significant decisions such as choice of college major.
Bouwmeester, S., Vermunt, J. K., & Sijtsma, K. (2007). Development and individual differences in transitive reasoning: A fuzzy trace theory approach. Developmental Review, 27(1), 41-74. doi: https://doi.org/10.1016/j.dr.2006.08.001
Braine, M. D. (1964). Development of a grasp of transitivity of length: A reply to Smedslund. Child Development, 35(3), 799-810. doi: https://doi.org/10.2307/1126505
Brannon, E.M. (2002). The development of ordinal numerical knowledge in infancy. Cognition, 83, 223–240. doi: https://doi.org/10.1016/S0010-0277(02)00005-7
Bryant, P.E., & Trabasso, T. (1971). Transitive inferences and memory in young children. Nature, 232(5311), 456–458.
Chalmers, M., & McGonigle, B. (1984). Are children any more logical than monkeys on the five-term series problem? Journal of Experimental Child Psychology, 37(2), 355-377. doi: https://doi.org/10.1016/0022-0965(84)90009-2
de Hevia, M.D., & Spelke, E.S. (2010). Number-space mapping in human infants. Psychological Science, 21 (5), 653–660. doi: https://doi.org/10.1177/0956797610366091
Gazes, R. P., Hampton, R. R., & Lourenco, S. F. (2017). Transitive inference of social dominance by human infants. Developmental Science, 20(2). doi: https://doi.org/10.1111/desc.12367
Hamlin, J.K., & Wynn, K. (2011). Young infants prefer prosocial to antisocial others. Cognitive Development, 26(1), 30–39. doi: https://doi.org/10.1016/j.cogdev.2010.09.001
Inhelder, B., and Piaget, J. (1958). The growth of logical thinking: From childhood to adolescence (A. Parsons and S. Milgram, Trans.). NY NY: Basic Books. (Original Work published in 1955).
Markovits, H., & Dumas, C. (1999). Developmental patterns in the understanding of social and physical transitivity. Journal of Experimental Child Psychology, 73(2), 95-114. doi: https://doi.org/10.1006/jecp.1999.2496
Piaget, J. (1952). The child’s conception of number. London: Routledge & Kegan Paul Ltd.
Piaget, J. (1959). The language and thought of the child: Selected works vol. 5. Routledge, London.
Odom, R. D., & Coon, R. C. (1967). A questionnaire approach to transitivity in children. Psychonomic Science, 9(6), 305-306. doi: https://doi.org/10.3758/BF03327818
Wright, B. C., & Smailes, J. (2015). Factors and processes in children’s transitive deductions. Journal of Cognitive Psychology, 27(8), 967-978. doi: https://doi.org/10.1080/20445911.2015.1063641
Dr. Chris Drew is the founder of the Helpful Professor. He holds a PhD in education and has published over 20 articles in scholarly journals. He is the former editor of the Journal of Learning Development in Higher Education. [Image Descriptor: Photo of Chris]