The Art of Logical Reasoning: Understanding Deduction and Induction

1. logic THE SCIENCE OF REASONING

2. What is logic? • Logic is the Study of the principles and concepts of good reasoning. • Yes, there is such a thing as bad reasoning. • Logicians are not interested in why people reason the way they do. • Logicians are interested in the principles of reasoning.

3. Biology, is the study of life and living organisms. • Life and living organisms are the subject matter of biology. • The subject matter of logic is the study of ARGUMENTS. • An argument is reasoning in support of a point. For example: • US presidents always have been men. • Men by nature are competitive and belligerent, women compassionate and caring. • But competition and war have led us to many issues—to mention a few examples, to social and economic inequality, the neglect of public schools, and a disastrous health system. • If we want to make “America great again”, we should allow a woman to become president. • Therefore, in the next elections you should vote for a woman.

4. ARGUMENT DEFINITION • A group of statements in support of a conclusion. A paper, A Speech, A book, An article, An oral presentation, can be arguments. But to be arguments they require a conclusion. Conclusion: the main point or thesis.

5. Good argument • A Good argument has • True premises. • And a conclusion that follows from the premises eithernecessarily or probably. NECESSARILY and PROBABLY imply 2 different ways to argue.

6. The 2 ways are 1. DEDUCTION2. INDUCTION If the conclusion follows necessarily, the argument is DEDUCTIVE. If it follows probably, it is INDUCTIVE. • If you are in Brooklyn, you are in the US. • You are in Brooklyn. • It follows that… How does it follow? You are in the US

7. 1. deduction

8. DEDUCTIVE means that given the premises, the conclusion either follows necessarilyor it doesn’t follow at all—all or nothing! • Given the premises, if the conclusion does follow necessarily, a deductive argument is called is VALID. • Given the premises, if the conclusion does not follow at all, a deductive argument is called INVALID.

9. Note that an argument can be deductively VALID, even if it contains one or more false premises. For example: • If you live in Brooklyn you are in South America. • You live in Brooklyn. • Therefore, you are in South America. • In this case we call the argument UNSOUND.

10. A Valid ARGUMENT that has ALL TRUE PREMISES is called sound • All physical objects occupy space. • My book is a physical object. • Therefore, my book occupies space.

11. What if the conclusion does not fallow at all?Then the argument is invalidfor example: • If you leave your car out on the street and it rains, your car get wet. • Your car is wet. • It follows that it rained.

12. Careful!An argument can be invalid even if it contains all true premises. • To be president of the US one must be 35 or older. • Trump is older than 35. • Therefore, Trump is the president of the US.

13. 2. induction

14. Often arguments are intended to support the conclusion with a matter of probability: • Every time I come to your house, your cat rubs against me. • I’m coming to your house later today. • Therefore, your cat will rub against me. • Here the premises are not meant to support the conclusion necessarily. • Given the premises, conclusion does not follow necessarily. • Does it not follow at all? No! • So how does it follow? • It follows probably.

15. INDUCTION • Given the premises, an argument is called INDUCTIVE just if the conclusion follows probably. There are three kinds of inductive arguments: • Inductive Generalizations. • Arguments from Analogy. • Causal Arguments.

16. 1. INDUCTIVE GENERALIZATION • To move from a sample to a general conclusion about a population. • This desk is brown. • That desk is brown. • Therefore, all desks are brown.

17. 2. ARGUMENTS FROMANALOGY • Using an analogy between two or more things (also people, events, etc.) in order to support a conclusion about one of them. • This book is boring. • That book has the same author and same plot. • Therefore, that book must also be boring. • Earth has oxygen. • Planets that have oxygen might have life. • Europa has oxygen. • Therefore, there might be life on Europa.

18. 3. CAUSAL ARGUMENT • An arguments that contains a causal statement as either a premise or the conclusion. • Exercising a lot makes you fit. • You are very fit. • Therefore, you exercise a lot. • Fever is caused by demonic possession. • You have fever. • Therefore, you are possessed by demons.

19. INDUCTIVE Conclusion always follows probably DEDUCTIVE Conclusion follows necessarily or not at all INVALID: Given the premises, the conclusion does not follow at all—even if the premises are true. NOT A GOOD ARGUMENT UNCOGENT: True premises but weak. Weak means given the premises, the conclusion is not likely to be true. NOT A GOOD ARGUMENT VALID: Given the premises, the conclusion follows necessarily. UNCOGENT: False premises but strong. Strong means given the premises, the conclusion is very likely to be true. NOT A GOOD ARGUMENT SOUND: A deductively valid argument with all true premises THIS IS A GOOD ARGUMENT COGENT: Strong plus all true premises. THIS IS A GOOD ARGUMENT UNSOUND: A deductively valid argument with at least one false premise. NOT A GOOD ARGUMENT

20. TEST YOUR KNOWLEDGE

21. Most bachelors are happy. • Mark is a bachelor. • So, Mark is happy. • Oh no! Eating soap gives you stomach ache, Jennifer. • You are eating soap. • Your stomach will ache, Jennifer.

22. If I am eating, obviously I have food. • And I am eating. • Therefore I have food. • Whenever I turn the knob right the volume increases. • Therefore, turning the knob right causes the volume to increase.

23. If it rains my car is wet. • It rains. • Therefore my car is wet. All bachelors are males. Joe is a male. Therefore, Joe is a bachelor.

24. All humans are mortal. • Nancy is human. • Therefore, Nancy is mortal. 40% of all CUNY teachers we interviewed are underpaid. So, all CUNY teachers are underpaid.

25. The moon is made of peanut butter. • Peanut butter can be eaten. • Therefore, the moon can be eaten. If you are anywhere in France, then you are in Europe. You are in Paris. Paris is in France. Therefore, you are in Europe.

26. All US presidents have been men. • Research shows that US citizens are tired of male presidents and are ready for a female president. • Consequently, next US president will be a female. Fordham is in Brooklyn. Brooklyn is in NYS. Therefore, Brooklyn was built before Manhattan.

27. VALID VS INVALID Distinguish valid from invalid arguments and sound from unsound All dogs have 4 legs. All tables have 4 legs. Therefore, all dogs are tables.

28. All actors are robots. Tom Cruise is an actor. Therefore, Tom Cruise is a robot.

29. People from England speak English. Marc is from England. Therefore, Marc speaks English.

30. The president of the US must be born in the US. Donald Trump was born in the US. Therefore, Donald Trump is the president of the US.

31. All actors are robots. Tom Cruise is a robot. Therefore,Tom Cruise is an actor.

32. British people speak English. Marc speaks English. Therefore, Marc is British.

33. INVALID! The Pope speaks 13 languages. This man speaks 13 languages. Therefore, this man is the Pope.

34. INVALID! 1. All female TV hosts are successful TV hosts. 2. Oprah Winfrey is a successful TV host. 3. Therefore, Oprah Winfrey is a female TV host.

35. Premise 1 says that all females who are TV hosts are successful. So, if you are a female, and you are a TV host, you are successful. However, premise 1 does not assert that ALL TV hosts are females. This implies that there are successful male TV hosts too. Now, what does premise 2 assert? It asserts that Oprah Winfrey is a successful TV host. But it does not say anything about her sex. I know what you’re thinking: “But I know Oprah! She is a female TV host.” Granted, but the question is NOT who Oprah in fact is. Once again, the question is whether the premises imply the conclusion. It is a matter of relationship between premises and conclusion, and not between reality and conclusion.

36. VALID! BUT UNSOUND Students who received an F pass the course. Philippa received an F. Therefore Philippa passed the course.

37. INVALID If you are playing soccer, you have a soccer ball. You have a soccer ball. Therefore you are playing soccer.

38. INVALID All trees are plants. This is a plant. Therefore, this is a tree.

39. INVALID All plants are green. Trees are green. Therefore, trees are plants.

40. INVALID All things with 3 sides are triangles. Some things with 3 angles are triangles. Therefore, some things with 3 angles are things with 3 sides.

41. Let’s change terms: Things with 3 sides become ants Triangles become insects Things with 3 angles become things that fly

42. All ants are insects. • Some things that fly are insects. • Therefore, some things that fly are ants. INVALID

43. VALID BUT UNSOUND All dogs are cats. Fluffy is a dog. Therefore, Fluffy is a cat.

44. ? VALID But is it sound? Whatever begins to exist has a cause. The universe began to exist. Therefore, the universe has a cause.

45. ARGUMENT OR NOT?

46. ARGUMENT? Research has shown that people who do at least 30 minutes a day of vigorous exercise reduce their risk of heart disease and some forms of cancer. It would be wise for you to begin a daily program of exercise.

47. PREMISE 1: Research has shown that people who do at least 30 minutes a day of vigorous exercise reduce their risk of heart disease and some forms of cancer. CONCLUSION: It would be wise for you to begin a daily program of exercise.

48. In 2014, we introduced a new textbook in the Logic course taught at SFC, and we negotiated with the publishers to have a discount for the students. It was agreed with the publisher that the textbook would cost $95, but the book was listed at $130. When students complained about the price, we called the publisher. Eventually, we understood what happened: the textbook was more expensive because the person with whom we had made an agreement on a reduced price had left the company, and the new employee wasn’t aware of the agreement. Therefore, at some point, the agreement was lost, and that’s why we didn’t get the discount.

49. NOT AN ARGUMENT!

50. Paying chief executives hundreds of times more than ordinary employees creates economic inequality, which contributes to political instability, violent crime, and reduced life expectancy. Whatever contributes to political instability, violent crime, and reduced life expectancy is not morally justified. So, there is no moral justification for chief executives being paid hundreds of times more than ordinary employees.