2.4 Deductive Reasoning

1. 2.4 Deductive Reasoning HW: Lesson 2.4/ 1 – 10, 13

2. Familiarize you with the deductive reasoning process Learn the relationship between inductive and deductive reasoning Objectives

3. Problem Solving • Logic – The science of correct reasoning. • Reasoning – The drawing of inferences or conclusions from known or assumed facts. • When solving a problem, • 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.

4. 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.

5. Deductive reasoning is the process of reasoning logically from given statements to a conclusion.

6. Examples of Inductive Reasoning • Every quiz has been easy. Therefore, the test will be easy. • The teacher used PowerPoint in the last few classes. Therefore, the teacher will use PowerPoint tomorrow. • Every fall there have been hurricanes in the tropics. Therefore, there will be hurricanes in the tropics this coming fall.

7. The catalog states that all entering freshmen must take a mathematics placement test. Example of Deductive Reasoning An Example: You are an entering freshman. Conclusion: You will have to take a mathematics placement test.

8. x 60◦ Inductive or Deductive Reasoning? Geometry example… Triangle sum property – the sum of the angles of any triangle is always 180 degrees. Therefore, angle x = 30° Deductive Reasoning – conclusion is based on a property

9. 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

13. 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.

14. 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!

15. Who is known for using Deductive Reasoning?

16. 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

17. However… • Deductive reasoning may not be the most accurate way of solving a problem, because we all know that assumptions can be wrong.

18. 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

19. 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.

20. Science Using laws and rules to make assumptions The law of gravity means everything that goes up must come downI threw a baseball in the airThat means the baseball must come down. Social Studies To be elected President you must obtain at least 270 electoral votesGeorge Bush won 287 electoral votesTherefore George Bush is the President.

21. What professions do you think commonly use Deductive Reasoning? Doctors Journalists Lawyers Detectives And many more...

22. 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.

28. 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.

29. 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.

30. 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.

31. If M is the midpoint of a segment, then it divides the segment into two congruent segments. M is the midpoint of . You can conclude that M divides into two congruent segments, or

32. If an angle is obtuse, then it is not acute. is not obtuse. The converse of the assumption is not true, so It is not possible to conclude that is acute.

33. 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.

34. bulbs 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?

35. 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.

37. 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.

38. 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.

39. 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!

40. 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.

41. 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.

42. 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

43. 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”

44. 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.

45. 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.