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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 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 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 mABC<90, then sophomores take driver ABC is an acute angle. education at her school. mABC = 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 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 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 pr 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 pr 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 pr 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?
