Laws of Logic

Using arguments that have logical order

Review Terms

• Counterexample• Conditional Statement• Hypothesis• Conclusion

• If Osama Bin Laden dies, the US troops will come home.

• The troops came back home.

• Conclusion: Osama is dead.

1. Identify the conditional statement.

2. Identify the hypothesis and conclusion.

3. Is this true?

4. Justify your answer.

• If you eat too much ice cream, you will get sick.

• You’re sick.

• Conclusion: You had too much ice cream.

1. Identify the conditional statement.

2. Identify the hypothesis and conclusion.

3. Is this true?

4. Justify your answer.

Law of Detachment

• If pq is a true conditional statement and p (hypothesis) is true, then q (conclusion) is true.

p q true

p true

q true

Law of Detachment

If you save a penny then you earn a penny.

Julio saves a penny.

Therefore, Julio earns a penny

Why is this argument valid? Because it follows the law of detachment.

p q true

p true

q true

p: save a penny

q: earn a penny

TRUE conditional statement

TRUE hypothesis

You can conclude that the conclusion is TRUE

Given

Given

Conclusion

Law of Detachment

If you pay attention then you will learn.

Mara pays attention.

Therefore, Mara will learn.

Why is this argument valid? Because it follows the law of detachment.

p q true

p true

q true

p: pay attention

q: will learn

TRUE conditional statement

TRUE hypothesis

You can conclude that the conclusion is TRUE

Given

Given

Conclusion

Law of Detachment

If you exercise then you will be healthy.

Tony is healthy.

Therefore, Tony exercises.

Why is this argument invalid? Because it does not follow the law of detachment.

p q true

p true

q true

p: exercise

q: healthy

TRUE

This is not the hypothesisYou cannot conclude that this statement is TRUE

Given

Given

Conclusion

Which argument is valid?

Vertical angles are congruent

A B

Therefore A and B are vertical angles

Vertical angles are congruent

A and B are vertical angles

Therefore A B

Invalid Valid

Which argument is valid?

If two lines are perpendicular, then they intersect to form right angles.

Lines l and m intersect to form right angles

Therefore, line l is perpendicular to line m

If two lines are perpendicular, then they intersect to form right angles.

Line l is perpendicular to line m

Therefore, lines l and m intersect to form right angles

Invalid Valid

Which argument is valid?

The menu says that apple pie a la mode is served with ice cream.

Laura ordered apple pie a la mode.

Therefore, she was served ice cream.

The menu says that apple pie a la mode is served with ice cream.

Laura ordered ice cream.

Therefore, she was served apple pie a la mode.

Valid Invalid

What can you conclude?Linear pairs are adjacent angles that measure 180°.

A and B are linear pairs

Therefore, A and B are adjacent angles and they measure 180°.

Linear pairs are adjacent angles that measure 180°.

A and B are adjacent angles

Therefore, This argument does not follow the Law of Detachment so I can not make a conclusion

Law of Detachment

Sarah knows that all sophomores take driver education at her school. Hank is taking driver education. So Hank is a sophomore.

If mABC<90, then ABC is an acute angle. mABC = 42 degrees. So ABC is an acute angle

1. Explain why this argument is valid/not valid.

2. Justify your answer.

3. What do you need to change to make a valid argument not valid and the not valid one valid.

ClosureMichael knows that if he does not do his chores in the morning, he will not be allowed to play video games later the same day. Michael does not play video games on Friday afternoon. So Michael did not do his chores on Friday morning.

If two angles are vertical, then they are congruent. ABC and DBE are vertical. So ABC and DBE are congruent.

Which statement is valid and which is not valid. Justify your answer.

Law of Syllogism

• If pq and qr are true conditional then pr is true.

How is the conclusion of the first conditional statement related to the hypothesis of the second conditional statement.

p q true

q r true

pr true

You can conclude this statement is TRUE

Conclusion

Law of Syllogism

If the sun is shining then it is a beautiful day.

If it is a beautiful day, then we will have a picnic.

Therefore if the sun is shining then we will have a picnic.

TRUE conditional statements

Given

Given

p q true

q r true

pr true

p: sun is shining

q: beautiful day

r: have a picnic

Why is this argument valid? Because it follows the law of syllogism.

You can conclude this statement is TRUE

Conclusion

Law of Syllogism

If you take algebra 1 then you will take geometry.

If you take geometry, then you will take algebra 2.

Therefore if you take algebra 1 then you will take algebra 2.

TRUE conditional statements

Given

Given

p q true

q r true

pr true

p: algebra 1

q: geometry

r: algebra 2

Why is this argument valid? Because it follows the law of syllogism.

You can conclude this statement is TRUE

Conclusion

Law of SyllogismIf you get the new job then you will be able to take the Mertolink.

If you take the Metrolink, then you will not have to buy a new car.

If you don’t have to buy an new car then you will not need to get insurance.

If you get the new job then you will not need to get

insurance.

TRUE conditional statements

Given

Given

p q true

q r true

r s true

p s true

p: get the new job

q: take the Metrolink

r: do not have to buy a new car

s: do not need insurance

Why is this argument valid? Because it follows the law of syllogism.

Given

Not the correct conclusion

Conclusion

Law of Syllogism

If 2 is acute then 3 is obtuse.

If 3 is obtuse, then 4 is acute.

Therefore if you 4 is acute then 2 is acute

TRUE conditional statements

Given

Given

p q true

q r true

pr true

p: 2 is acute

q: B is obtuse

r: 4 is acute

Why is this argument invalid?

Because it doesn’t follow the law of syllogism.

Which argument is valid?

Valid Invalid

If the two lines are parallel then the lines do not intersect.

If the lines don’t intersect, then no angles are formed.

Therefore if the two lines are parallel then the no angles are formed.

If the two lines are parallel then the lines do not intersect.

If the lines don’t intersect, then we will no angles are formed.

Therefore if no angles are formed then the two lines are parallel

Which argument is valid?

Valid Invalid

If we visit Hong Kong, then we will eat well.

If we eat well, then we will walk a lot.

If we visit Hong Kong then we will walk a lot.

If we visit Hong Kong, then we will eat well.

If we visit Hong Kong, we will walk a lot.

If we eat well then we will walk a lot.

Which argument is valid?

Invalid Valid

If we visit Disneyland then we will see Mickey Mouse.

If we visit Disneyland then we will get on Space Mountain.

If we get on Space Mountain then we will have fun.

If we see Mickey Mouse then we get on Space Mountain.

If we visit Disneyland then we will see Mickey Mouse.

If we visit Disneyland then we will get on Space Mountain.

If we get on Space Mountain then we will have fun.

If we visit Disneyland then we will have fun.

• If Don is going, then Eve is going.

• Ben is not going to the party.

• If Al is going then, Ben is going.

• If Carla is going, then Don is going

• Al or Carla is going to the party.

• Is Ben going to the party?

• Is Al going to the party?• Is Carla going to the

party?• Is Don going to the

party?• Is Eve going to the

party?

Who is going to the Party?Using the Law of Detachment and the Law of Syllogism