Logical Reasoning

1. Logical Reasoning • Deductive reasoning • Inductive reasoning

2. Deductive Reasoning • Reasoning from the general to the specific • For example, start with a general statement: All cars have tires. • You can apply this general statement to specific instances and deduce that a Ford Escort, a Toyota Camry, and a Mercedes Benz must have tires.

3. Common deductive reasoning problems • Series problems • Syllogisms

4. Series problems • review series of statements • arrive at a conclusion not contained in any single statement • For example: • Robin is funnier than Billy • Billy is funnier than Sinbad • Whoopi is funnier than Billy • Q: Is Whoopi funnier than Sinbad

5. Syllogisms • Present two general premises that must be combined to see if a particular conclusion is true

6. Syllogism Example • All Intro to Psychology students love their instructor. • You are all Intro to Psychology students. • Must you love your instructor?

7. Syllogism Example • All chefs are violinists. • Mary is a chef. • Is Mary a violinist?

8. Ways to solve syllogisms • Mental model theories • Pragmatic reasoning theories

9. Psych- ology Psych- ology Psych- ology Bi- ology Bi- ology Bi- ology Bi- ology Mental models theories • To solve a syllogism, you might visualize the statements • All Intro to Psychology students love their instructor. • You are all Intro to Psychology students. • Must you love your instructor? YES! YES! YES!

10. Psych- ology Psych- ology Psych- ology Bi- ology Bi- ology Bi- ology Bi- ology Mental models theories • All Intro to Psychology students love their instructor. • You are all Biology students. • Must you love your instructor? NO! NO! NO!

11. Psych- ology Psych- ology Psych- ology Bi- ology Bi- ology Bi- ology Bi- ology Mental models theories • Syllogisms that are easy to visualize are more readily solved than more abstract syllogisms

12. Mental model theories • To solve a syllogism, you might visualize the statements • Syllogisms that are easy to visualize are more readily solved than more abstract syllogisms

13. Pragmatic reasoning theories • Solve syllogisms by applying information to pre-existing schemas • Problem difficulty related to importance of problem to our lives and survival as a species • More relevant = easier to solve

14. Inductive reasoning • Reasoning from the specific to the general

15. Inductive reasoning • 181614???? 12 10 • Rule? Decrease by 2 • Q: Why inductive reasoning? • Answer: Take SPECIFIC numbers (i.e. 18,16,14) and come up with a GENERAL rule (i.e. decrease by 2)

16. Inductive Reasoning • Sherlock Holmes is perhaps a better example of INDUCTIVE reasoning than deductive reasoning • He takes specific clues and comes up with a general theory

17. Inductive reasoning problems • 7 8 1617 ???? 25 26 • 4 8 5 10 ?? ?? ?? 7 14 11 • 720 120 24 ?? ?? ?? 6 2 1

18. Inductive reasoning problems • 5 10 15 ?? ?? ?? ?? ?? ?? ?? ?? 20 25 30 35 40 45 50 55 • Rule? • Increase by five WRONG!!!!! • What is the correct rule? • Any increasing number • - the next number could be 87 or 62 or 1,000,006 • Why did everyone guess the wrong rule?

19. Confirmation bias • Only search for information confirming one’s hypothesis • Example: reading newspaper columnists who agree with our point of view and avoiding those who don’t

20. Chris story • Chris is 6’7”, 300 pounds, has 12 tattoos, was a champion pro wrestler, owns nine pit bulls and has been arrested for beating a man with a chain. • Is Chris more likely to be a man or a woman? • A motorcycle gang member or a priest? • How did you make your decision?

21. Steve story • Steve is meek and tidy, has a passion for detail, is helpful to people, but has little real interest in people or real-world issues. • Is Steve more likely to be a librarian or a salesperson? • How did you come to your answer?

22. Representativeness • Judge probability of an event based on how it matches a prototype • Can be good • But can also lead to errors • Most will overuse representativeness • i.e. Steve’s description fits our vision of a librarian

23. Most will underuse base rates • Base rate - probability that an event will occur or fall into a certain category • Did you stop to consider that there are a lot more salespeople in the world than librarians? • By sheer statistics, there is a greatly likelihood that Steve is a salesperson. • But very few take this into account

24. Guess the probabilities • How many people die each year from: • Heart disease? • Floods? • Plane crashes? • Asthma? • Tornados? Stop

25. Availability heuristic • Judge probability of an event by how easy you can recall previous occurrences of that event. • Most will overestimate deaths from natural disasters because disasters are frequently on TV • Most will underestimate deaths from asthma because they don’t make the local news

26. Word probabilities • Is the letter “k” most likely to occur in the first position of a word or the third position? • Answer: “k” is 2-3 times more likely to be in the third position • Why does this occur?

27. Class demonstration • Name words starting with “k” • Name words with the letter “k” in the third position

28. Availability heuristic • Because it is easier to recall words starting with “k” , people overestimate the number of words starting with “k”

29. Finish the sequence problems • 30 24 18 ?? ?? ?? 12 6 0 • Rule? • Decrease by six • 1 3 2 4 ?? ?? ?? ?? 3 5 4 6 • Rule? • Increase by two, decrease by 1

30. Finish the sequence problems • 2 3 1012???????????????????????? ?? ???? ?? ?? ?? ???? ?? 13 20 21 22 29 30 31 32 39 200 299 300 301 201 302 399 2000 • Rule? • Increasing numbers starting with the letter “t”

31. Chess problem • Two grandmasters played five games of chess. Each won the same number of games and lost the same number of games. There were no draws in any of the games. How could this be so? • Solution: They didn’t play against each other.

32. Bar problem • A man walked into a bar and asked for a drink. The man behind the bar pulled out a gun and shot the man. Why should that be so? • Solution: The man behind the bar wasn’t a bartender. He was a robber.

33. Bar problem # 2 • A man who wanted a drink walked into a bar. Before he could say a word he was knocked unconscious. Why? • Solution: He walked into an iron bar, not a drinking establishment.

34. Nine dots problem • Without lifting your pencil or re-tracing any line, draw four straight lines that connect all nine dots

35. Answer to nine dots problem

36. Metal Set • Q: Why couldn’t you solve the previous problems? • A: Mental set - a well-established habit of perception or thought

37. Strategies for solving problems • 1. Break mental sets

38. Number problem mental set 13 20 21 22 29 30 • 2 3 1012???????????????????????? ?? ???? ?? ?? ?? ???? ?? 31 32 200 299 300 301 39 201 302 399 2000 • Most people get stuck in the same rhythm • Only view problems in terms of math formulas • Need to break out of this mental set to solve the problem

39. Nine dots mental set • Most people will not draw lines that extend from the square formed by the nine dots • To solve the problem, you have to break your mental set

40. Mounting candle problem • Using only the objects present on the right, attach the candle to the bulletin board in such a way that the candle can be lit and will burn properly

41. Answer to candle problem • Most people do not think of using the box for anything other than it’s normal use (to hold the tacks) • To solve the problem, you have to overcome functional fixedness

42. Functional fixedness • type of mental set • inability to see an object as having a function other than its usual one

43. Strategies for solving problems • 1. Break mental sets • break functional fixedness • 2. Find useful analogy

44. Find useful analogy • Compare unknown problem to a situation you are more familiar with

45. Strategies for solving problems • 1. Break mental sets • 2. Find useful analogy • 3. Represent information efficiently • 4. Find shortcuts (use heuristics)

46. Two general classes of rules for problem solving • 1. Algorithms • 2. Heuristics

47. Two general classes of rules for problem solving • Algorithms - things the vice-president might say • Algorithms - rules that, if followed correctly, will eventually solve the problem

48. An algorithm example • Problem: List all the words in the English language that start with the letter “q” • If using an algorithm, would have to go through every single possible letter combination and determine if it were a word • i.e. is “qa” a word; is “qb” a word etc. • This would take a very long time • Instead, what rule could you use to eliminate these steps?

49. Rules for “q” problem • Skip ahead and assume the second letter is a “u” • Assume the third letter has to be a vowel • These types of rules are called heuristics

50. Heuristics • Any rule that allows one to reduce the number of operations that are tried in problem solving • a.k.a rules of thumb or shortcuts • Another common heuristic: • Problem: List all the numbers from 1-100,000 that are evenly divisible by 5 • Answer: Rather than divide each and every number, you would use the rule: Any number ending in 0 or 5 is evenly divisible by 5.