The process of solving mathematical problems is similar to developing strategies to overcome obstacles encountered in real life. When we face a “problem,” we explore different solution paths and focus on the essential aspects by disregarding unnecessary information. This process requires reasoning. The probability of each choice resulting in a positive or negative outcome depends on the context, conditions, and factors associated with that choice. Therefore, to find the correct solution, we must evaluate various possibilities and make a logical selection.

Logic and Reasoning

Reasoning is the process of deriving a “conclusion” from a given premise (a prior knowledge). In reasoning, the statement or propositions serving as evidence or justification are referred to as “premises,” and the proposition derived and justified as a result is called the “conclusion.” Logic, in this context, studies the relationship between premises and conclusions in reasoning.

Logic examines the formal structure of reasoning, independent of the content of the propositions. For this reason, it is referred to as “formal logic.” The truth of propositions, in terms of content, is a matter of “knowledge,” while logic concerns the validity of reasoning.

Validity refers to whether the conclusion necessarily follows from the premises in a reasoning process. In logic, reasoning is divided into three main types:

1. Induction (Inductive Reasoning): Derives general conclusions from specific observations.
2. Deduction (Deductive Reasoning): Reaches specific conclusions based on general statements.
3. Analogy (Analogical Reasoning): Draws conclusions based on similarities between two situations.

In daily life, we use both inductive reasoning and deductive reasoning to make decisions and solve problems.

Reasoning

The likelihood of each choice leading to a positive or negative outcome depends on the context, conditions, and factors involved in that choice. Evaluating these probabilities is a critical process for making logical and correct decisions.

Logical thinking involves analyzing a given situation and evaluating all possibilities and outcomes. This process requires making inferences, understanding cause-and-effect relationships, and considering alternative paths. In short, the reasoning process is actively using logical thinking skills to make sound decisions.

Inductive Reasoning

Inductive reasoning is a process where a general conclusion is derived from specific examples or observations. For example:

• “All the swans I have seen are white. Therefore, all swans are white.”
• “The weather has been sunny for three months. Therefore, it will be sunny tomorrow.”

This method allows us to make generalizations based on observations. However, the validity of the generalization depends on the representativeness and scope of the examples.

Question 1: What do you think the continuation of the numbers will be?

1 + 2 + 3 + 4 + 5 + 6 + …

2 + 4 + 6 + 8 + 10 + 12 + …

Question 2: After adding two odd numbers, if we subtract the result from the sum, we get an even number. Explain this with inductive reasoning.

Deductive Reasoning

Deductive reasoning involves drawing a conclusion based on general statements that have already been accepted as true. It relies on general rules rather than specific examples. For example:

• “All vehicles must adhere to speed limits in traffic. Therefore, this car must also adhere to the speed limit.”

Example 1: Consider any number. Multiply it by 3, then add 6, and divide the result by 3. Afterward, subtract the original number. What is the result?