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,
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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
An Example:
The catalog states that all entering freshmen
must take a mathematics placement test.
You are an entering freshman.
Conclusion: You will have to take a
mathematics placement test.
Inductive or Deductive Reasoning?
Geometry example…
x
Triangle sum property –
the sum of the angles of any
triangle is always 180 degrees.
Therefore, angle x = 30°
60◦
Deductive Reasoning – conclusion is based on a property
Inductive or Deductive Reasoning?
Geometry example…
What comes next?
Is there a rule?
Colored triangle rotating 90° CW in
the corners of the square
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 Holmes
Sherlock Holmes would use deductive reasoning to help
solve crimes.
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.
Andrew Ault
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 Engineers
George is not from M.I.T.
Therefore George is not an Engineer
Everybody from Texas is a cowboy
Scott is from Texas
Scott is a cowboy
How Can Deductive Reasoning Be Applied
In School?
Math
Scott 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.
English
When I see ‘like’ in a sentence, I deduce that it is a simile.
Science
Using laws and rules to make assumptions
The law of gravity means everything that goes up must
come down
I threw a baseball in the air
That means the baseball must come down.
Social Studies
To be elected President you must obtain at least 270
electoral votes
George Bush won 287 electoral votes
Therefore 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 AB into
two congruent segments, or AM  MB.
If an angle is obtuse, then it is not acute.
XYZ is not obtuse.
The converse of the assumption is not
true, so It is not possible to conclude
that XYZ is acute.
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?
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 example
All 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 Reasoning
Examples:
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 parts
2 parts are called the hypothesis & conclusion
Can 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.