Algebra III
Lesson 7
Inductive and Deductive Reasoning
– Logic –
The Contrapositive - Converse and Inverse
Inductive and Deductive Reasoning
Inductive Reasoning
- Generalize to find a rule
- Take data and try to start a pattern
- A good guess that can easily be wrong
Deductive Reasoning
Logical Reasoning
- Use an existing rule
- A bad rule can give false results
- Proper use of rules
- Truth or falseness are not under consideration
Logic
Syllogism
Premise
Categorical Propositions
- The process of using logic from two premises
- A statement can be true or false
- Sort things into categories
Universal affirmatives or Major Premise
States that everything in a group has a certain property
Minor Premise
Relates to one thing in the group
1) All men are mortal ……… (Major Premise)
2) Aristotle is a man ………. (Minor Premise)
Therefore ( ), Aristotle is mortal…….. (Valid Conclusion)
Following these steps is called an argument
A valid conclusion does not guarantee truth
Logic follows the major premise exactly from beginning
to end, not the other way around
1) All frogs are green
2) Frank is a frog
Frank is green
(Valid)
1) All frogs are green
2) Frank is green
Frank is a frog
(Invalid – followed major premise backwards)
1) If it rains, I will go to town
2) It did not rain
I did not go to town
(Invalid – only know what happens if it
Rains, not if it doesn’t)
[Might go to town anyway…]
Example 7.1
Is the following argument a valid argument?
Why or why not?
All Normal dogs have four legs
That dog has four legs
That dog is normal
Invalid –
Followed major premise backwards, the dog could have a missing tail.
Example 7.2
Is the following argument a valid argument?
Why or why not?
All boys are good
That child is a good child
That child is a boy
Invalid –
Went backwards on Major Premise
Example 7.3
Is the following argument a valid argument?
Why or why not?
All chickens have three legs
Henny Penny is a chicken
Henny Penny has three legs
ha d
Valid
How about Truth?
Is the conclusion true?
The Contrapositive
A premise has two parts: Hypothesis and Conclusion
Hypothesis begins with ‘IF’
Conclusion begins with ‘THEN’
When a premise doesn’t have if or then in it, it can be rewritten
For example: Rabbits are fast runners.
Rewrite as: If it is a rabbit,
then it is a fast runner.
Premises only work one way
If
then
Negation
Put a not or non- in the if and/or the then part of the premise
For example:
If it is a rabbit,
If it is not a rabbit,
If it is a non-rabbit,
Negations of the
original if statement
To make a contrapositive
1st) Flip the if and then portions of the premise
2nd) Negate both parts
For example:
If it is a rabbit, then it is a fast runner.
1st) If it is a fast runner, then it is a rabbit.
2nd) If it is not a fast runner, then it is not a rabbit.
If the original statement is true, then its contrapositive is also true.
Example 7.4
Is the following argument valid?
All nonathletes are vegetarians.
Jim is a nonvegetarian._______
Jim is an athlete.
Since we don’t know about nonvegetarians, make the
contrapositive of the major premise.
All nonvegetarins are athletes.
Jim is a nonvegetarian.______
Jim is an athlete.
This is a valid argument, so the original argument is also valid.
Converse and Inverse
If normal dog then four legs
If rabbit then fast
If P then Q
P is always the original/starting hypothesis,
Q is always the original/starting conclusion.
‘~’ is the shorthand symbol for negation.
Given P → Q is the major premise:
Then:
~Q → ~P is the contrapositive
Q → P is the converse
~P → ~Q is the inverse
No matter what the truth of the original statement, the
converse and the inverse have no guarantee of validity
or truth.
“If-and-only-if” statements (iff)
These work in both directions
Definitions also work in both directions
Measure of 90°
P
↔
↔
Right angle
Q
Practice
a)
Write as if-then statement
Elephants are large animals
If the animal is an elephant, then it is a large animal
b)
Write the contrapositive
If an animal is green, then it is not a seal
If it is a seal, then it is not green
c) Write the contrapositive of the major premise of this syllogism to
help determine whether the following argument is valid
If the car is not moving, then the motor is off
The motor is on
.
The car is moving
To find out about having the motor on, take the
contrapositive of the major premise
If the motor is on, the car is moving
The motor is on
.
The car is moving
Valid