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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 pq 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
Given If you save a penny then you earn a penny.TRUE conditional
statement
Given Julio saves a penny. TRUE hypothesis
You can conclude that the
Conclusion Therefore, Julio earns a penny
conclusion is TRUE
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
Law of Detachment
Given If you pay attention then you will learn. TRUE conditional
statement
Given Mara pays attention. TRUE hypothesis
You can conclude that the
ConclusionTherefore, Mara will learn.
conclusion is TRUE
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
Law of Detachment
Given If you exercise then you will be healthy.
TRUE
Given Tony is healthy.
This is not the hypothesis
You cannot conclude that
ConclusionTherefore, Tony exercises.
this statement is TRUE
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
Which argument is valid?
Vertical angles are congruent
Vertical angles are congruent
A  B
A and B are vertical angles
Therefore 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.
If two lines are perpendicular,
then they intersect to form right
angles.
Lines l and m intersect to form
right angles
Line l is perpendicular to line m
Therefore, lines l and m intersect
Therefore, line l is perpendicular to form right angles
to line m
Invalid
Valid
Which argument is valid?
The menu says that apple pie a
la mode is served with ice
cream.
The menu says that apple pie a
la mode is served with ice
cream.
Laura ordered apple pie a la
mode.
Laura ordered ice cream.
Therefore, she was served ice
cream.
Valid
Therefore, she was served apple
pie a la mode.
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
If mABC<90, then
sophomores take driver
ABC is an acute angle.
education at her school.
mABC = 42 degrees.
Hank is taking driver
So ABC is an acute
education. So Hank is a
angle
sophomore.
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.
Closure
Michael 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 pq and qr are true conditional then
pr 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
pr
true
Law of Syllogism
Given If the sun is shining then it is a beautiful day. TRUE
Given If it is a beautiful day, then we will have a
picnic.
conditional
statements
You can
Conclusion Therefore if the sun is shining then we will have conclude this
a picnic.
statement is
TRUE
Why is this argument valid? Because it follows the law of syllogism.
p q true
p: sun is shining
q r
q: beautiful day
pr
true
true
r: have a picnic
Law of Syllogism
Given If you take algebra 1 then you will take
geometry.
TRUE
conditional
statements
Given If you take geometry, then you will take algebra
You can
2.
conclude this
statement is
TRUE
Conclusion Therefore if you take algebra 1 then you will
take algebra 2.
Why is this argument valid? Because it follows the law of syllogism.
p q true
p: algebra 1
q r
q: geometry
pr
true
true
r: algebra 2
Law of Syllogism
Given
Given
If 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.
TRUE
conditional
statements
If you don’t have to buy an new car then you will not need
to get insurance.
You can conclude
Conclusion If you get the new job then you will not need to get this statement is
insurance.
TRUE
Given
Why is this argument valid? Because it follows the law of syllogism.
p q true
p: get the new job
q r
q: take the Metrolink
true
r  s true
p s true
r: do not have to buy a new car
s: do not need insurance
Law of Syllogism
Given If 2 is acute then  3 is obtuse.
TRUE conditional
Given If  3 is obtuse, then  4 is acute. statements
Conclusion Therefore if you  4 is acute then 2 is acute Not the correct
conclusion
Why is this argument invalid?
p q true
p: 2 is acute
q r
pr
q:  B is obtuse
true
true
r:  4 is acute
Because it doesn’t follow the
law of syllogism.
Which argument is valid?
If the two lines are parallel
then the lines do not intersect.
If the two lines are parallel
then the lines do not intersect.
If the lines don’t intersect, then
no angles are formed.
If the lines don’t intersect, then
we will no angles are formed.
Therefore if the two lines are
parallel then the no angles are
formed.
Therefore if no angles are
formed then the two lines are
parallel
Valid
Invalid
Which argument is valid?
If we visit Hong Kong, then we
will eat well.
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, we will
walk a lot.
If we visit Hong Kong then we
will walk a lot.
If we eat well then we will
walk a lot.
Valid
Invalid
Which argument is valid?
If we visit Disneyland then we will
see Mickey Mouse.
If we visit Disneyland then we will
see Mickey Mouse.
If we visit Disneyland then we will
get on Space Mountain.
If we visit Disneyland then we will
get on Space Mountain.
If we get on Space Mountain then we
will have fun.
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
have fun.
Invalid
Valid
Who is going to the Party?
Using the Law of Detachment and the Law of Syllogism
• 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?
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