2.4 Deductive Reasoning

HW: Lesson 2.4/ 1 – 10, 13

OBJECTIVES

• Familiarize you with the deductive reasoning process• Learn the relationship between inductive and deductive

reasoning

Problem Solving

• Logic – The science of correct reasoning.• Reasoning – The drawing of inferences or

conclusions from known or assumed facts.• When solving a problem,

• one must understand the question, • gather all pertinent facts, analyze the problem • i.e. compare with previous problems

(note similarities and differences), • perhaps use pictures or formulas to solve the

problem.

Deductive vs. Inductive Reasoning

• The difference:inductive reasoning uses patterns to arrive

at a conclusion (conjecture)

deductive reasoning uses facts, rules, definitions or properties to arrive at a conclusion.

Deductive reasoning is the process of reasoning logically

from given statements to a conclusion.

Examples of Inductive Reasoning

1) Every quiz has been easy. Therefore, the

test will be easy.2) The teacher used PowerPoint in the last

few classes. Therefore, the teacher will use PowerPoint tomorrow.

3) Every fall there have been hurricanes in the tropics. Therefore, there will be hurricanes in the tropics this coming fall.

Example of Deductive Reasoning

The catalog states that all entering freshmen must take a mathematics placement test.

Conclusion: You will have to take a mathematics placement test.

You are an entering freshman.

An Example:

Inductive or Deductive Reasoning?

Geometry example…

60◦

x Triangle sum property – the sum of the angles of

any triangle is always 180 degrees. Therefore, angle x = 30°

Deductive Reasoning – conclusion is based on a property

Inductive or Deductive Reasoning?

Geometry example…

What comes next?

Colored triangle rotating 90° CW in the corners of the square

Is there a rule?

Inductive Reasoning

Deductive Reasoning Is…

• Deductive reasoning is when you start from things you assume to be true, and draw conclusions that must be true if your assumptions are true.

For Example

All dogs have a tail.

Buddy is a dog.

Therefore Buddy has a tail.

Still don’t get it?• Man approaches to greet a new neighbor who is just moving into the house

next door and asks what he does for a living. • Neighbor 1: I am a professor at the University, I teach deductive reasoning. • Man: Deductive reasoning? What is that? • Neighbor 1: Let me give you an example. I see you have a dog house out

back. By that I deduce that you have a dog. • Man: That's right. • Neighbor 1: The fact that you have a dog, leads me to deduce that you have

a family. • Man: Right again. • Neighbor 1: Since you have a family I deduce that you have a wife. • Man: Correct. • Neighbor 1: And since you have a wife, I can deduce that you are

heterosexual. • Man: Yup. • Neighbor 1: That is deductive reasoning!

Who is known for using Deductive Reasoning?

Sherlock HolmesSherlock Holmes would use deductive reasoning to help solve crimes.

Andrew Ault

Example:Sherlock Holmes could help solve a mystery by making inferences.

If Holmes saw a pack of cigarettes by a victim (the victim did not smoke), Holmes can make the assumption that the killer is a smoker.

However…

• Deductive reasoning may not be the most accurate way of solving a problem, because we all know that assumptions can be wrong.

Other faults of deductive reasoning

All Graduates of M.I.T. are EngineersGeorge is not from M.I.T.Therefore George is not an Engineer

Everybody from Texas is a cowboyScott is from TexasScott is a cowboy

How Can Deductive Reasoning Be Applied In School?

MathScott has a case of soda in his house since there are 13 cans of soda left I deduce that Scott has drank 11 cans of soda.

EnglishWhen I see ‘like’ in a sentence, I deduce that it is a simile.

ScienceUsing laws and rules to make assumptions

The law of gravity means everything that goes up must come downI threw a baseball in the airThat means the baseball must come down.

Social StudiesTo be elected President you must obtain at least 270 electoral votesGeorge Bush won 287 electoral votesTherefore George Bush is the President.

What professions do you think commonly use Deductive

Reasoning?

Why?• These professions tend to ask a lot of questions to try

to solve problems or to prove a point.• Often they would have to make assumptions to solve

problems.• They would use rules and widely accepted beliefs to

prove their argument.

For example:An attorney states that his client is innocent because the crime victim was hit by a car. Since his client does not have a license. He can deduce that his client is innocent.

An auto mechanic knows that if a car has a dead battery, the car will not start. A mechanic begins work on a car and finds the battery is dead. What conclusion will she make?

The mechanic can conclude that the car will not start.

If there is lightning, then it is not safe to be

out in the open. Marla sees lightning

from the soccer field.

It is not safe for Marla to be out in the

open.

If it is snowing, then the temperature is less than or equal to 32˚F.

The temperature is 20˚F.

The converse of the original assumption is not true, so

It is not possible to conclude that it is snowing.

If M is the midpoint of a segment, then

it divides the segment into two

congruent segments.

M is the midpoint of .AB

You can conclude that M divides into

two congruent segments, or

AB

.MBAM

If an angle is obtuse, then it is not acute.

is not obtuse.XYZ

The converse of the assumption is not true, so It is not possible to conclude that is acute.XYZ

If a road is icy, then driving conditions are hazardous.

Driving conditions are hazardous.

The converse of the assumption is not true, so It is not possible to conclude

that the road is icy.

There are three switches downstairs. Each

corresponds to one of the three light

bulbs in the attic. You can turn the

switches on and off and leave them in any

position. How would you identify which

switch corresponds to which light bulb, if

you are only allowed one trip upstairs?

bulbs

Answer

Turn one light switch one.Wait about 20 minutes, then turn it off.

Turn another light on, and proceed upstairs.One light should be on,

Two lights are off- however, one light is hot and one is cold.

Deductive Reasoning

• This method of reasoning produces results that are certain within the logical system being developed.

• It involves reaching a conclusion by using a formal structure based on a set of undefined terms and a set of accepted unproved axioms or premises.

• The conclusions are said to be proved and are called theorems.

Deductive Reasoning

• Deductive Reasoning – A type of logic in which one goes from a general statement to a specific instance.

• The classic exampleAll men are mortal. (major premise)Socrates is a man. (minor premise)Therefore, Socrates is mortal. (conclusion)

The above is an example of a syllogism.

Deductive Reasoning• Syllogism: An argument composed of two

statements or premises (the major and minor premises), followed by a conclusion.

• For any given set of premises, if the conclusion is guaranteed, the arguments is said to be valid.

• If the conclusion is not guaranteed (at least one instance in which the conclusion does not follow), the argument is said to be invalid.

• BE CARFEUL, DO NOT CONFUSE TRUTH WITH VALIDITY!

Deductive ReasoningExamples:1. All students eat pizza.

Claire is a student at CSULB.Therefore, Claire eats pizza.

2. All athletes work out in the gym. Barry Bonds is an athlete. Therefore, Barry Bonds works out in the gym.

Deductive Reasoning

3. All math teachers are over 7 feet tall. Mr. D. is a math teacher. Therefore, Mr. D is over 7 feet tall.• The argument is valid, but is certainly not true.

Examples

• No one who can afford health insurance is unemployed.

All politicians can afford health insurance. Therefore, no politician is unemployed.

VALID OR INVALID?????

The argument is valid

Conditional Statements

A logical statement with 2 parts2 parts are called the hypothesis & conclusionCan be written in “if-then” form;

such as, “If…, then…”

Hypothesis is the part after the word “If”Conclusion is the part after the word “then”

Underline the hypothesis & circle the conclusion.

If you are a brunette, then you have brown hair.

hypothesis conclusion

Vertical angles are congruent.

If there are 2 vertical angles, then they are congruent.If 2 angles are vertical, then they are congruent.

Converse

Switch the hypothesis & conclusion parts of a conditional statement.

Write the converse of “If you are a brunette, then you have brown

hair.”If you have brown hair, then you are a

brunette.