DEDUCTIVE REASONING

This section will hopefully be a refresher topic on deductive reasoning.  Of course, we cannot let you get away that easy!

Now that you have a good understanding of deductive reasoning, we will add some more applications for this concept. You will learn about the Law of Syllogism, which you may have also seen in a previous math class.

You would have learned it as the Transitive property. Don’t worry, even if you don’t remember that property, we will show you how it works.

What is deductive reasoning?

Answer :

The process of drawing logically certain conclusions by using an argument is known as deductive reasoning.

Deductive reasoning is a valid form of proof.

Two Laws of Deductive Reasoning

Law of Detachment :

An if-then statement is a form of deductive reasoning. 

We assume that if the "if" part is true, then, by the Law of Detachment, it automatically follows that the "then" part is always true.

Example :

If you take this medicine regularly, you will be recovered soon.

Law of Syllogism :

Law of Syllogism is nothing but the transitive property of logical statements.

That is,

If p → q and  q → r, then  p → r.

Example :

Two if-then statements :

1. If an object is a square, then it is a rhombus.

2. If an object is a rhombus, then it is an equilateral.

By Law of Syllogism, we have

1. If an object is a square, then it is a rhombus.

2. If an object is a rhombus, then it is an equilateral.

That is,

If an object is a square, then it is an equilateral

Solved Problems

Problem 1 :

Draw a logical conclusion from the given statements :

I can go to the party, if I clean my room.

I cannot go to the party.

Solution :

By Law of Detachment, we can have the following conclusion for the above statements.

I did not clean my room

Problem 2 :

What conclusion can be drawn :

If I do my homework, then I will get good grades.

If I get good grades, then my parents will be happy.

Solution :

By Law of Syllogism, we can have the following conclusion for the above statements.

If I do my homework, then my parents will be happy.

Problem 3 :

Write some conditional statements that can be made from the following true statements using the Law of Syllogism.

1. If a bird is the fastest bird on land, then it is the largest of all birds.

2. If a bird is the largest of all birds, then it is an ostrich.

3. If a bird is a bee hummingbird, then it is the smallest of all birds.

4. If a bird is the largest of all birds, then it is flightless.

5. If a bird is the smallest bird, then it has a nest the size of a walnut half-shell.

Solution :

Here are the conditional statements that use the Law of Syllogism.

a. If a bird is the fastest bird on land, then it is an ostrich. (Use 1 and 2.)

b. If a bird is a bee hummingbird, then it has a nest the size of a walnut half-shell. (Use 3 and 5.)

c. If a bird is the fastest bird on land, then it is flightless. (Use 1 and 4.)

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