1. Formal Logic
From ages 11 to 12, there gradually develops what Piaget called the formal operational stage. New capabilities at this stage, like deductive (if-then) reasoning and establishing abstract relationships, are generally mastered around ages 15 to 16.
As we saw, by the end of this stage, teenagers, like adults, can use both formal and abstract logic—but only if they have learned the language of logic (“if,” “then,” “therefore,” etc.) and have repeatedly put it to use. Under these circumstances, children learn to extrapolate and make generalizations based on real-life situations.
Thus, from ages 10 to 12, by stimulating children intellectually—urging them to reflect and establish lines of reasoning—they gradually become able to move beyond a situational logic based on action and observation onto a logic based on rules of deduction independent of the situation at hand.
This ability to manipulate abstract symbols consolidates by around age 15, provided that one has been versed in formal logic.
Here is an example of how formal logical faculties can be sparked:
A and B are two logical propositions, such that A is the opposite of B. From this, we may formally deduce (without reference to anything concrete) that the proposition P, which states “A or B,” is always true. There are no alternatives, so P fulfills all possibilities. We may also deduce that the proposition P1, “A and B,” is always false. Here, two contradictory propositions cannot both be true. If one is true, the other is false.
These formal operations require both a mature central nervous system and a mature cognitive system. But, since such examples of formal reasoning are detached from everyday life, they require deliberate practice. Even an adult who is out of practice can struggle with formal reasoning.
We can present these two rules of logic using more concrete examples, which makes formal reasoning at once more accessible and less intimidating. In concrete form, however, the reasoning will be less easily applied to new situations.
If proposition A is: “this salmon is farmed,” proposition B (the opposite of A) will be: “this salmon is not farmed.” B could also be expressed as: “this salmon is wild.” It is easy in this concrete context to see that P, “A or B,” is always true. A salmon must either be farmed or wild. It is also easy to conceive that P1, “A and B,” is always false because a salmon cannot be both farmed and wild.
Once they learn to abstract from concrete examples and express these rules in formal logic, children can form and manipulate logical notation and apply it to a multitude of situations.
How can we help children from age 13 and older improve their formal logical deduction skills?
We must start by working on these two rules through concrete examples like that of the salmon. After working through several examples, we can help children extract the logical rules behind them. This is the inductive phase: from concrete examples, we extract the common features and express them in a formal rule.
Next, it will be necessary to prove this rule solely by logical deduction. If we do not do this, we cannot be certain that the rule is valid in every context. Extracting the common features only results in rules which, at this stage, remain merely hypothetical. Only reasoning allows for the generalization of a rule.
Once students have mastered a collection of formal rules, they can be trained to recognize, within a problem or a given context, what rule is applicable. That is, they can take an initial claim (a hypothesis), apply a rule of deduction to it, and arrive at a conclusion.