Reasoning

1. Reasoning

2. Reasoning • What is reasoning? • The world typically does not give us complete information • Reasoning is the set of processes that enables us to go beyond the information given

3. What types of reasoning are there? • Validity vs. Truth • Valid argument: true premises guarantee a true conclusion • It does not necessarily correspond to the truth in the world • Deductive reasoning • Allows us to draw conclusions that must hold given a set of facts (premises) • Inductive reasoning • Allows us to expand on conclusions • Conclusions need not be true given premises • Category-based induction • Analogical reasoning • Mental models

4. The logic of the situation • You have tickets to the football game. Go Mean Green! • You agree to meet Bill and Mary at the corner of Fry and Hickory or at the seats • If you see Mary on the corner of Fry and Hickory, you expect to see Bill as well. • If you do not see either of them at the corner, you expect to see them at the seats when you get to the stadium. • The agreement has a logical form • (Bill AND Mary) will be located at corner OR • (Bill AND Mary) will be located at seats • AND and OR are logical operators • They have truth tables

5. The logic of the situation • Simple logical arguments • If you see Mary • Bill AND Mary • You expect to see Bill

6. Limits of logical reasoning • We are good at this kind of reasoning • We do it all the time • We can do it in novel situations • Are we good at all kinds of logical reasoning? • What are our limitations?

7. Conditional Reasoning • Modus Ponens • Modus Tollens

8. Conditional Reasoning • Each card has a letter on one side, and a number on the other • Which Cards must you turn over to test the rule: • If there is a vowel on one side of the card, then there is an odd number on the other side

9. Conditional Reasoning • Who do you have to check? • If you have a beer, then you must be 21 or older?

10. Conditional Reasoning • These cases are logically the same • Valid Arguments: If premises are true, conclusion must be true • Affirming the Antecedent • P Q • P • Q (Modus Ponens) • Denying the Consequent • P Q • NOT Q • NOT P (Modus Tollens)

11. Conditional Reasoning • Invalid Arguments: Conclusion need not be true, even if premises are true. • Affirming the Consequent • P  Q • Q • P • Denying the Antecedent • P  Q • NOT P • NOT Q • The ambiguity of if. In everyday language, sometimes implies a bidirectional relationship between P and Q (i.e. if and only if)

12. Logical thinking • Pure logic says that we should be able to reason about any content • The Ps and Qs in the argument could be anything • However, we are more likely to accept an argument when the conclusion is true (in the real world) whether it is valid or not • All professors are educators • Some educators are smart • Some professors are smart • This conclusion may be true • The argument is not valid • It is possible that the smart educators are not professors

13. Logical thinking • We are good with simple logical operators • AND, OR, NOT • Earlier we saw content effects • Wason selection task • With neutral content it is more difficult • With familiar content it is easier • Social schemas are easy to reason about and may be context dependent rather that • Cheng & Holyoak; Tooby & Cosmides • E.g. Permission: Some precondition must be filled in order to carry out some action • More complex argument forms can be difficult, especially in unfamiliar contexts • Why do we see these content effects? • Valid deductive arguments ensure that a conclusion is true if the premises are true • Truth cannot be determined with certainty, thus we must generally reason about content • We will look at how people reason about content later

14. Inductive Reasoning • Luci’s presentation!

15. Abductive Reasoning • Say what? • Another form of reasoning is provided by the philosopher C.S. Peirce • It essentially provides a means for coming up with rules based on new instances experiences • One way you might think of it is coming up with hypotheses based on new findings (whereas deduction would deal with outlining the consequences of a hypothesis and induction in testing the hypothesis) • Observation: the grass is wet • Explanation: it rained • The explanation is consistent with the domain of the problem

16. Abuduction • Deduction • Necessary inferences (if A leads to B and B leads to C, then A leads to C) • All balls in this urn are red • All balls in this particular random sample are taken from this urn • Therefore All balls in this particular random sample are red • Peirce regarded the major premise here as being the Rule, the minor premise as being the particular Case, and the conclusion as being the Result of the argument. • The argument is a piece of deduction (necessary inference): an argument from population to random sample.

17. Abuduction • Induction • Interchange the conclusion (the Result) with the major premise (the Rule). Argument becomes: • All balls in this particular random sample are red • All balls in this particular random sample are taken from this urn • Therefore, All balls in this urn are red • Here is an argument from sample to population, and this is what Peirce understood to be the core meaning of induction: argument from random sample to population

18. Abuduction • Abduction • New argument: Interchange the conclusion (the Result) with the minor premise (the Case) • Argument becomes: All balls in this urn are red • All balls in this particular random sample are red • Therefore, All balls in this particular random sample are taken from this urn. • This is nothing at all like an argument from population to sample or an argument from sample to population: it is a form of probable argument different from both deduction and induction • Would later see these as three aspects of the scientific method

19. Scientific reasoning • Scientific reasoning • Combination of reasoning abilities • Hypothesis testing • Generate an explanation for some phenomenon • Develop an experiment to test the hypothesis • Seek disconfirming evidence • How good are people at this type of reasoning? • How good are scientists at living up to this ideal?

20. Hypothesis Testing • Deductive side (conditional reasoning) • If the null hypothesis is true, this data would not occur • The data has occurred • The null hypothesis is false • This is true by denying the consequent (modus tollens) • Unfortunately this is not how hypothesis testing takes place • If the null hypothesis is true, this data would be unlikely • The data has occurred • The null hypothesis is false • The problem is that we make the first statement probabilistic, and that changes everything

21. If a person is an American, then he is not a member of Congress FALSE This person is a member of Congress Therefore, he is not an American This is a valid argument but untrue as the first premise is false If a person is an American, then he is probably not a member of Congress TRUE This person is a member of Congress Therefore, he is not an American This is the form of hypothesis testing we undertake, and is logically incorrect Hypothesis Testing

22. Hypothesis testing • Induction • Take a sample, calculate a statistic • Generalize to the population • Problem: often no real reason to believe the population statistic is a constant • Example: though the transformed score is of course a mean of 100 IQ, IQ raw scores have been improving over the past couple decades • Begs the question, to what are we generalizing? Just this population at this time?

23. Hypothesis testing • People tend to have a confirmation bias • We seek confirming evidence • Scientists also show a confirmation bias • They tend to be more critical of evidence that is inconsistent with their beliefs. • This always may not be a bad thing (Koehler) • Wason 246 task • You are told to find a rule that generates “correct” three number sequences. • You are told that 2-4-6 is a “correct” sequence • You search for the rule by testing as many sequences as you want until you are confident you know the rule

24. Hypothesis testing • Confirmation bias • Many people initially assume the rule is “Sequences increasing by 2” • They try sequences like “4-6-8” and “13-15-17” • These are sequences that would confirm their hypothesis • Few people try sequences that would disconfirm their hypothesis (e.g., “1-2-3” or “3-2-1”) • The actual rule is “Any increasing sequence” • Few people find the correct rule

25. Hypothesis testing • Scientists ignore base rates (prior research) • Bayes theorem allows for incorporating prior probabilities to give a (posterior) probability about a hypothesis • Yet most of social science does not use Bayesian methods • Some do not realize that the end of their scientific efforts is a probability about data, not a hypothesis • Not p(H|D) • But p(D|H)

26. MC’s experience at Research and Statistical Support • People (students and faculty) come in with: • No clear hypothesis to test • Lack of knowledge regarding the methods that would allow a hypothesis to be tested • Heavy reliance on prescribed ‘rules’ which do little to aid their reasoning about the problem • Vague notion as to which population they are generalizing to • And a host of other issues…

27. MC’s Suggestions for Having Fun with Science • Have clear ideas • Regarding concepts (operational definitions), their implications, and the coherency of hypotheses regarding them • Sounds easy but is probably the hardest part and the source of most problems • Do not ignore prior efforts • Sorry to break it to you, but much has been done in your area of research • Don’t be afraid to explore • Engage your natural curiosity (try new methods and really investigate your data) • Think causally • Every method is an investigation of a causal model, what’s yours? • Remember the big picture • Your research should speak well beyond its specific results (esoterism ≠ progress)

28. Importance of Content • Analogy and Similarity • How do we use past experience? • What are analogies? • Structural alignment • Similarity

29. What to do... • How do you decide what to buy? • Use your past experience • How do you figure out which experience is relevant? • Using prior knowledge • Use of prior knowledge is guided by similarity • How can we study this process? • Studying pairs of items • Study perceptions of similarity when all information is available

30. Contrast model • Tversky (1977) • Had people list features of concepts • Had other people rate the similarity of concepts • Compared the feature lists • Similarity increases with common features, similarity decreases with distinctive features • Similarity ratings were positively related to the number of common features • Similarity ratings were negatively related to the number of distinctive features

31. Analogy • Often, things being compared are not very similar. • Atom vs. Solar system • Analogies preserve relations • The Atom and the Solar System have similar relations among their parts. • The Atom • cause( • greater(charge(nucleus) • charge(electron)), • revolve(electron,nucleus)) • The Solar System • cause( • greater(mass(Sun) • mass(planet)), • revolve(planet,Sun)) • The attributes of the objects are not similar. • The nucleus is not hot, the planets are not small etc.

32. Structure mapping • Structured representations • Relations connect the objects • Items are placed in correspondence when they play the same role in a matching relational system

33. Analogical Inference • Can make inferences about target domain • Inferences based on correspondences between the base and target • Allows us to learn from experience

34. Types of similarity

35. Focus on alignable differences • Gentner & Markman (1994) • Ss given 40 word pairs • 20 highly similar, 20 highly dissimilar • Hotel-Motel • Magazine-Kitten • List one difference for as many pairs as possible in 5 minutes • More differences listed for similar pairs than dissimilar pairs • Reflects that alignable differences are easier to find for similar pairs than for dissimilar pairs

36. Similarity and cognition • Similarity enables us to use background knowledge • Recognize how a new case is like an old one • Structure mapping/structural alignment • Relations are important in similarity comparisons • Commonalities and Alignable differences are key • Nonalignable differences are less important • Differences are easy to find for similar things • Structural alignment affects cognitive processing

37. Reasoning and Mental Models • Mental models • Intuitive Theories and Naïve physics

38. Mental Models and Intuitive Theories • Mental models allow us to reason about devices • Kind of like scripts and schemas discussed earlier • People often have causal information about the way things work • Used to allow us to get through the world • Information may be flawed • Three types of mental models • Logical mental models • Analogical mental models • Causal models

39. Logical and Analogical Models • Logical mental models • Used to solve logic problems • Johnson-Laird • Contain “empty” symbols that are manipulated • All Archers are Bankers • No Bankers are Chemists • ? • Useful primarily for logic puzzles • Analogical mental models • Sometimes we understand one device by analogy to another • Electricity and water flow • Voltage <--> Water pressure • Current <--> Flow rate • Resistance <--> Width of pipe

40. Causal Models • Causal models allow us to explain and understand the world around us • Note that it is not exactly clear what may be determined what a cause is, and that is a separate question from how we come to a determination of a causal relationship • White 1990, Ideas about Causation in Philosophy and Psychology • Nevertheless our (often flawed) notions of causal relationships can have profound effects on our ability to reason and understand

41. Intuitive Theories • Naïve physics • What would happen to a ball shot through this pipe? • People often respond by assuming curvilinear momentum • McCloskey and Proffitt • Even happens if they carry out an action

42. Intuitive Theories • Why do we err? • Our naïve physics matches our observations • The world has friction, and so there are unseen forces that act in opposition to seen forces • Our naïve physics is often accurate for things we can do with our bodies • Only when we create larger machines do the differences become important • Should not be a surprise • Newtonian physics is only a few hundred years old • Aristotelian mechanics is closer to our daily experience

43. How deep are our models? • Shallowness of explanation • Keil • People believe they understand more than they do • Asked college students about devices • Toilet, Car ignition, Bicycle derailleur • Said they understood devices, but could not actually explain them • Why does this happen? • When we know how to use an object and it is familiar, we believe we know how it works

44. Summary • Mental models • Logical mental models • Analogical mental models • Causal mental models • Naïve physics • Physical beliefs sometimes diverge from truth • Sufficient to get us around the world • Scientific reasoning • People generate pretty good tests • Often show a confirmation bias