2.1 Deductive and Inductive Reasoning

1. 2.1 Deductive and Inductive Reasoning

2. 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.

3. 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.

4. 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.

5. 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.

6. 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°

7. Inductive or Deductive Reasoning? Geometry example…

11. 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.

12. 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.

13. 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!

14. Deductive Reasoning Examples: • 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.

15. Deductive Reasoning Examples: • All students eat pizza. Claire is a student at wmhs. Therefore, Claire eats pizza. 2. All athletes work out in the gym. LebronJames is an athlete. Therefore, Lebron James works out in the gym.

16. Deductive Reasoning 3. All math teachers are over 7 feet tall. Mr. Pis a math teacher. Therefore, Mr. P is over 7 feet tall. • The argument is valid, but is certainly not true. • The above examples are of the form If p, then q. (major premise) x is p. (minor premise) Therefore, x is q. (conclusion)

17. Venn Diagrams Venn Diagram: A diagram consisting of various overlapping figures contained in a rectangle called the universe. U This is an example of all A are B. (If A, then B.) B A

18. Venn Diagrams This is an example of some A are B. (At least one A is B.) The yellow oval is A, the blue oval is B.

19. Example • Construct a Venn Diagram to determine the validity of the given argument. #14 All smiling cats talk. The Cheshire Cat smiles. Therefore, the Cheshire Cat talks. VALID OR INVALID???

20. ExampleValid argument; x is Cheshire Cat Things that talk Smiling cats x

21. Examples • #6 No one who can afford health insurance is unemployed. All politicians can afford health insurance. Therefore, no politician is unemployed. VALID OR INVALID?????

22. Examples X=politician. The argument is valid. People who can afford Health Care. Politicians X Unemployed

23. Example • #16 Some professors wear glasses. Mr. Einstein wears glasses. Therefore, Mr. Einstein is a professor. Let the yellow oval be professors, and the blue oval be glass wearers. Then x (Mr. Einstein) is in the blue oval, but not in the overlapping region. The argument is invalid.

24. Inductive Reasoning Inductive Reasoning, involves going from a series of specific cases to a general statement. The conclusion in an inductive argument is never guaranteed. Example: What is the next number in the sequence 6, 13, 20, 27,… There is more than one correct answer.

25. Inductive Reasoning • Here’s the sequence again 6, 13, 20, 27,… • Look at the difference of each term. • 13 – 6 = 7, 20 – 13 = 7, 27 – 20 = 7 • Thus the next term is 34, because 34 – 27 = 7. • However what if the sequence represents the dates. Then the next number could be 3 (31 days in a month). • The next number could be 4 (30 day month) • Or it could be 5 (29 day month – Feb. Leap year) • Or even 6 (28 day month – Feb.)

26. All bats are mammals. All mammals are warm-blooded. So, all bats are warm-blooded. All arguments are deductive or inductive. Deductive arguments are arguments in which the conclusion is claimed or intended to follow necessarily from the premises. Inductive arguments are arguments in which the conclusion is claimed or intended to follow probably from the premises. Is the argument above deductive or inductive?

27. All bats are mammals. All mammals are warm-blooded. So, all bats are warm-blooded. Deductive. If the premises are true, the conclusion, logically, must also be true.

28. Kristin is a law student. Most law students own laptops. So, probably Kristin owns a laptop. In the example above, the word probably shows that the argument is inductive.

29. No Texans are architects. No architects are Democrats. So, no Texans are Democrats.

30. Either Kurt voted in the last election, or he didn't. Only citizens can vote. Kurt is not, and has never been, a citizen. So, Kurt didn't vote in the last election. Arguments by elimination are arguments that seek to logically rule out various possibilities until only a single possibility remains. Arguments of this type are always deductive.

31. Tess: Are there any good Italian restaurants in town? Don: Yeah, Luigi's is pretty good. I've had their Neapolitan rigatoni, their lasagne col pesto, and their mushroom ravioli. I don't think you can go wrong with any of their pasta dishes. Based on what you've learned, is this argument deductive or inductive? How can you tell?

32. Don: Yeah, Luigi's is pretty good. I've had their Neapolitan rigatoni, their lasagne col pesto, and their mushroom ravioli. I don't think you can go wrong with any of their pasta dishes. Inductive.

33. I wonder if I have enough cash to buy my psychology textbook as well as my biology and history textbooks. Let's see, I have $200. My biology textbook costs $65 and my history textbook costs $52. My psychology textbook costs $60. With taxes, that should come to about $190. Yep, I have enough. Is this argument deductive or inductive? How can you tell?

34. Mother: Don't give Billy that brownie. It contains walnuts, and I think Billy is allergic to walnuts. Last week he ate some oatmeal cookies with walnuts and he broke out in a severe rash. Father: Billy isn't allergic to walnuts. Don't you remember he ate some walnut fudge ice cream at Melissa's birthday party last spring? He didn't have any allergic reaction then. Is the father's argument deductive or inductive? How can you tell?