Concepts & Flawed Reasoning
Concepts & Flawed Reasoning
2. Concepts and Flawed Reasoning
Extension vs. Intension
One idea in formal logic that can be valuable to learn at this age has to do with how concepts are defined. For very young children, categories or concepts are defined according to how they are encountered in everyday life. For example, the general concept of color is determined by all the examples of colors children have come across or imagined. The concept covers all these different experiences. This is called the concept’s “extension.”
But it is important that children from the age of around 13 start to learn to define concepts not merely according to their extension, but in a formal, scientific manner.
For example, instead of using a definition drawn from experience, students can explain that a color is a perception that our eye, linked to the brain, produces when an electromagnetic wave of a given frequency hits our retina. This definition according to the formal, internal qualities of the concept is called the concept’s “intension.”
Definition by intension is more complicated, but it allows for the use of the concept in formal reasoning. Therefore, definition by intension gears the child’s mind towards higher-level abstract reasoning.
For example, if we have to determine whether or not a given entity is a color or not, the intensional definition will offer us formal criteria for making a judgment.
Here’s another example. The prime numbers can be defined formally by intension: they are “the numbers that are only divisible by themselves and one.” If we were to learn only the extension of the term “prime number,” on the other hand, we would only have a list of the numbers that we know are prime.
It is clear that if we only have a this definition by extension and we encounter a new, very large number— higher than the largest number on the list we’ve learned—we will have no criteria for knowing whether it’s prime. But if we have the formal definition by intension, we will, with the help of a calculator, be able to determine whether it is only divisible by one and itself and, therefore, prime.
When we are young, we learn about the world through definitions by extension during the course of our interactions with objects and other people. Our brain defines concepts by extension and then extracts the common features to produce a working definition.
But these definitions are subjective since they depend on our history of encounters with relevant examples. Thus, all of the concepts we have created do not match other people’s concepts precisely, despite being identically named. They depend on the particular experiences we have had.
Yet, towards the ages of 13 to 15, with mathematical and formal logic, it becomes possible to define concepts by intension and, therefore, to share objective meaning with others. Teenagers can enter a world of shared and precise meanings. This is a prerequisite for the application of precise and formal critical thinking—for what good is critiquing the logic of others if I do not share their definitions of the concepts they use in their reasoning?
The formal approach for children aged 13 and up should, then be twofold: formalize the definition of the concepts used and formalize the logical deduction itself. This comes with practice and enhances both children’s capacity to communicate and their critical faculties.
Case Study 1
Recognizing Flawed Reasoning
As has been discussed in previous sections, developing critical reasoning requires more than simply knowing how to reason formally and contextually. It is also necessary to learn how to recognize flaws in the reasoning of other people who may wish to convince us of their way of thinking, either for narcissistic reasons or to lead us to act to their own advantage.
Such flaws can occur on several levels:
Erroneous rules of logic, leading to false reasoning based on reliable hypotheses.
False hypotheses (starting points for reasoning): even if the reasoning is valid, the conclusion may be false. Certain politicians use this strategy very frequently.
Using a formal rule in a situation to which it does not apply. This often occurs in over-simplified mathematical modeling of complex material, for example when an essay in the humanities is interpreted using only the tools of formal logic.
These three types of flaws can be worked into family discussions, with the goal of training children to counter weak or manipulative lines of argument. School should not be too heavily relied upon to provide this kind of practice for your children. Already between the ages of 13 and 15, they are able to construct brilliant lines of reasoning, which will prevent them from being tricked by manipulative or intellectually limited people.
Case Study 2