1183 LearnersLast updated on 11 September 2025
Law of Syllogism
Syllogism is a method of reasoning used to draw conclusions from given statements. It is used to draw conclusions from the given statements. The law of syllogism is the logical reasoning pattern that helps to make conclusions from two statements. Now let’s learn more about syllogism, its structure, types, and more.
What is the Law of Syllogism in Geometry?
In geometry, the law of syllogism is used in logical reasoning to draw conclusions from given statements. The word syllogism means deduction or inference in Greek. It is like a chain rule and similar to the transitive property; that is, if a = b and b = c, then a = c.
According to the law of syllogism in geometry, if two conditional statements are true, then:
- If p -> q (if p, then q)
- If q -> r (if q, then r)
- Therefore, p -> r (if p, then r)
What is the Structure of a Syllogism?
The law of syllogism is the fundamental principle in logical reasoning that allows us to conclude from two conditional statements. It has three parts: the first two are premises, and the last one is the conclusion. The conditional statement that follows the word “IF” is the hypothesis, and the inference follows after the word “THEN”.
The syllogism follows the pattern,
- Statement 1: If P, then Q
- Statement 2: If Q, then R
- Statement 3: If P, then R
Here, statements 1 and 2 are the premises. If both the premises are true, then the conclusion (statement 3) is true.
What are the Uses of the Law of Syllogism to Draw a Conclusion
Using the law of syllogism, we analyze the following statements:
P: If a triangle is equilateral, then its angles are 60°
Q: In a triangle, if all angles are 60°, then it is an equiangular triangle.
Since both conditions describe equivalent properties in Euclidean geometry, the conclusion depends on how the definitions are applied. If interpreted differently, the syllogism may not hold.
What are the 3 Types Of Syllogisms?
Syllogism has two statements, a major and a minor premise. While the major premise represents a general statement, the minor premise applies it to a specific case.
There are three types of syllogism:
- Conditional syllogism
- Categorical syllogism
- Disjunctive syllogism
- Conditional Syllogism: If statement A is true, then statement B is true
- Categorical Syllogism: If all A are in category C, and B is in A, then B is also in C.
- Disjunctive Syllogism: If statement A is true, then B is not true
Real-world applications of the Law of Syllogism
The law of syllogism is used in real-world situations to make decisions. These are some of its applications:
- Research: By linking observations to established theories, researchers use syllogisms.
- Geometry Proofs: The law of syllogism is used to prove mathematical relationships, mainly in geometry.
- AI Decisions: In AI, we use this law to make decisions.
Common Mistakes and How to Avoid Them in Law of Syllogism
Students often make errors when applying the law of syllogism.Here are a few common mistakes and ways to avoid them
Mistake 1
Confusing the Premises and Conclusion Structure
Confusing premises with conclusion structure is common among students. So, it is important to label the premise and conclusion before the analysis.
Mistake 2
Ignoring the Conditional Form
The conditional statement is the statement with if-then, and the law of syllogism is based on it. But overlooking these forms can lead to misapplications. So first identify the conditional statements by looking for the words “if” and “then” in the sentence.
Mistake 3
Applying the Law of Syllogism to Invalid Cases
Applying the law of syllogism to invalid statements is common among students. Ensure that the consequence of the first premise matches the condition of the second premise before drawing a conclusion.
Mistake 4
Remember: If A -> B does not mean B -> A, unless it is explicitly stated as ‘if and only if
Students often mistakenly use the converse of a conditional statement. So remember that a conditional statement only indicates one direction unless it’s stated as “if and only if.”
Mistake 5
Incorrectly Negating Statements When Forming a Conclusion.
Some students mistakenly negate statements when drawing conclusions, which can break the logical flow of reasoning. So, it is important to recognize the law of syllogism, which deals with direct relationships, and unnecessary negation can disrupt this flow.
Solved Examples of Law of Syllogism
problem 1
If a number is divisible by 6, then it is divisible by 3. If a number is divisible by 3, then it is an integer. What can we conclude?
Therefore, if a number is divisible by 6, it must be an integer.
Explanation
Using the law of syllogism, we can link the two conditionals.
If the number is divisible by 6, it is divisible by 3, and as it is divisible by 3, it is an integer.
So, it can be concluded that if a number is divisible by 6, then it is an integer.
problem 2
If a country is in South America, then it is in the Southern Hemisphere. If a country is in the Southern Hemisphere, then it experiences summer in December. What can we conclude?
If a country is in South America, then it experiences summer in December.
Explanation
The first statement says that any South American country lies in the Southern Hemisphere.
The second statement tells us that all locations in the Southern Hemisphere have summer in December.
Therefore, every country in South America experiences summer in December.
problem 3
If a company increases its advertising, then more people will learn about its products. If more people learn about its products, then the company’s sales will increase. What can we conclude?
If a company increases its advertising, then its sales will increase
Explanation
Using the law of syllogism, we connect the conditionals: increasing advertising leads to greater product awareness, and increased awareness leads to higher sales.
Hence, the act of increasing advertising implies that sales will increase.
problem 4
Statement 1: If a number is even, then it is divisible by 2. Statement 2: If a number is divisible by 2, then it is not an odd number. What conclusion can be drawn?
Thus, an even number cannot be odd.
Explanation
Here, the first statement confirms that even numbers are divisible by 2.
The second statement tells us that any number divisible by 2 cannot be odd.
Thus, if a number is even, it logically follows that it is not an odd number.
problem 5
If you run a red light, then you break the law. If you break the law, then you may get a fine. What can we conclude?
If you run a red light, then you may get a fine.
Explanation
Applying the law of syllogism, running a red light leads to breaking the law, and breaking the law leads to the possibility of a fine.
Thus, running a red light may result in a fine.
FAQs on Law of Syllogism
1.What is the law of syllogism?
2.What are the essential components of a syllogism?
3.What is the role of the middle term in a syllogism?
4.What are the types of syllogism?
5.In which academic fields is the law of syllogism most useful?
Explore More data
![Important Math Links Icon]()
Previous to Law of Syllogism
Jaipreet Kour Wazir
About the Author
Jaipreet Kour Wazir is a data wizard with over 5 years of expertise in simplifying complex data concepts. From crunching numbers to crafting insightful visualizations, she turns raw data into compelling stories. Her journey from analytics to education ref
Fun Fact
: She compares datasets to puzzle games—the more you play with them, the clearer the picture becomes!
