# Law of Syllogism

Law of Syllogism
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Imagine you are at the ticket counter of your favorite gaming store.

You are third in the queue.

Each transaction with a customer takes about 2 minutes.

You have only 10 minutes to get your ticket before the counter closes.

What can you infer from the above situation? Can you draw any conclusion?

You may be very happy that there is a fair chance that you can get your ticket.

While drawing this conclusion, you must have used the "Law of syllogism."

Those logical inferences that you draw are outcomes of the law of syllogism.

Even in a game of chess, you use several logical conclusions for taking the next steps in the game.

For example, if I defend my king what happens to the knight right up there!

Let us look at the law of syllogism in more detail and understand the definition of the law of syllogism and its examples in real life.

## Lesson Plan

 1 Law of Syllogism in Geometry 2 Thinking Out of the Box! 3 Solved Examples on Law Of Syllogism 4 Challenging Questions on Law Of Syllogism 5 Interactive Questions on Law Of Syllogism

## Law of Syllogism in Geometry

### Law of Syllogism Definition

The word "Syllogism" has a Greek origin and it means deduction or inference.

Syllogism refers to drawing inferences from given prepositions or sentences.

The Law of Syllogism is actually a part of deductive reasoning where we arrive at conclusions by logical reasoning.

It is similar to the transitive property: if a = b and b= c, then a=c.

It is like a chain rule.

### Law of Syllogism Example

Statement 1: If it is a Monday, I have school.

Statement 2: If I have school, I have my math class.

The conclusion that we can draw from the above two statements is, "If it is a Monday, then I have math class."

This law of syllogism is a wonderful tool for proving many mathematical statements, especially in geometry.

## Structure of a Syllogism

In the rule of syllogism, there are three parts involved.

Each of these parts is called a conditional argument.

The hypothesis is the conditional statement that follows after the word if.

The inference follows after the word then

To represent each phrase of the conditional statement, a letter is used.

The pattern looks like this:

Statement 1: If P, then Q.

Statement 2: If Q, then R.

Statement 3: If P, then R.

Statements 1 and 2 are called the premises of the given argument.

If they are true, then the correct inference must be statement 3.

## Using the Law of Syllogism to Draw a Conclusion

Let us look at this geometry problem.

Draw a conclusion from the following true statements using the Law of Syllogism.

P: If a quadrilateral is a square, then it has four right angles.

Q: If a quadrilateral has four right angles, then it is a rectangle.

Here, statement P is true but statement Q is not true.

So, even though there is an immediate inference that a square is a rectangle, it is not valid as P is true, but not Q.

## What Are the 3 Types Of Syllogisms?

Syllogisms are arguments usually with two statements (or premises).

Major premise: a point in general.

Minor premise: a particular argument.

The conclusion is based on both statements.

There are 3 main types of syllogisms. They are as follows.

Conditional Syllogism: If A is true, then B is true (If A, then B).

Categorical Syllogism: If A is in C, then B is in C.

Disjunctive Syllogism: If A is true, B is not true (A or B).

Now that we know what syllogism is about, let us try some examples to understand it better.

Think Tank

Choose the correct option using the law of syllogism:

Statement 1: All bats are mammals.

Statement 2: No birds are bats.

Conclusions:

a)     No birds are mammals.

b)     Some birds are not mammals.

c)     No bats are birds.

d)     All mammals are bats.

## Solved Examples

 Example 1

Help John draw a conclusion using the law of syllogism.

Statement 1: If a number ends in 0, then it is divisible by 10.

Statement 2: If a number is divisible by 10, then it is divisible by 5.

Solution

Let P be the statement "The number ends in 0"; let Q be the statement "It is divisible by 10"; and let R be the statement "It is divisible by 5."

Then (1) and (2) can be re-written as:

1) If P, then Q.

2) If Q, then R.

Thus, by the Law of Syllogism, we can infer:

3) If P, then R.

That means, if a number ends in 0, then it is divisible by 5.

 $$\therefore$$ If a number ends in 0, then it is divisible by 5.
 Example 2

Draw a conclusion using the law of syllogism.

All bikes have wheels. I ride a bike.

Solution

This scenario belongs to a categorical syllogism.

Major Premise: All bikes have wheels.

Minor Premise: I ride a bike.

So, by the law of syllogism, we can conclude that my bike has wheels.

 $$\therefore$$ Conclusion is - My bike has wheels.
 Example 3

Noah had a conversation with his friend.

Noah: All bunnies are cute.

Friend: My cousin's pet is also cute.

Therefore, the pet is a bunny.

Do you think the conclusion is valid?

Solution

These statements do not fall in any specific category of syllogism, so there is a high chance that we may end up in fallacy.

Major Premise: All bunnies are cute.

Minor Premise: My cousin's pet is also cute.

Conclusion: The pet is a bunny.

Not every cute pet is a bunny.

 $$\therefore$$ The conclusion is not valid.
 Example 4

Help Harry draw a conclusion using the law of syllogism.

$$\angle A$$ and $$\angle C$$ are equal.
$$\angle B$$ and $$\angle C$$ are equal.

Solution

So, the statements can be considered as:

P:  $$\angle A = \angle C$$

Q:  $$\angle B = \angle C$$

R:  $$\angle A = \angle B$$

Then (1) and (2) can be re-written as:

1) If P, then Q.

2) If Q, then R.

So, by the Law of Syllogism, we can infer:

3) If P, then R.

Thus, $$\angle A = \angle B = \angle C$$

 $$\therefore \angle A = \angle B = \angle C$$
Challenging Questions
1. Choose the valid conclusion using the law of syllogism:

Statements:

• All technicians are villagers.
• No villager is a doctor.
• All doctors are managers.

Conclusions:

1. No technician is a manager.
2. All villagers being managers is a possibility.

## Interactive Questions

Here are a few activities for you to practice.

## Let's Summarize

The mini-lesson targeted the fascinating concept of the law of syllogism. The math journey around the law of syllogism starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

### 1. What is the law of detachment?

The law of detachment helps to arrive at a new valid conclusion from the given statements.
1) If P, then Q
2) P

By the Law of Detachment, we can conclude that Q is valid.

For example,

1) If you are a bird, then you live in a nest.

2) You are a bird.

Let P be the statement, "You are a bird"; let Q be the statement, "You live in a nest".

By the Law of Detachment, we can conclude that Q is valid.

$$\therefore$$ You live in a nest.

### 2. What is the Law of Contrapositive?

The negation and inversion of the original statement which conveys the same meaning is called the contrapositive.

Interchange the hypothesis and the conclusion of the inverse statement to form the contrapositive of the given statement.

The law of contraposition states that the given statement is valid if and only if its contrapositive is true.

### 3. What is the Law of Converse?

Interchange the hypothesis and the conclusion in order to form the converse of the given statement.

For example,

Statement: If two triangles are congruent, then their corresponding angles are equal.

Converse: If the corresponding angles of the two triangles are equal, then the triangles are congruent.

Note that, the converse of a statement need not hold good in every case.

### 4. What are the three types of syllogism?

There are 3 main types of syllogisms. They are as follows.

Conditional Syllogism: If A is true, then B is true (If A, then B).

Categorical Syllogism: If A is in C, then B is in C.

Disjunctive Syllogism: If A is true, B is not true (A or B).

### 5. What is the pattern of a syllogism?

In the rule of syllogism, there are three conditional arguments.

The hypothesis is the conditional statement that follows after the word if.

The inference follows after the word then.

The pattern looks like this:

Major Premise: If P, then Q.

Minor Premise: If Q, then R.

Conclusion: If P, then R.

### 6. What is the purpose of syllogism?

The law of syllogism is especially used in proving geometrical statements.

It is also used to derive logical conclusions from the given statements (or premises).

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