REASONING AS PROBLEM SOLVING

1. REASONING AS PROBLEM SOLVING given a set of facts (premises), • DEDUCTIVE REASONING: • what, if any, conclusions necessarily follow? • INDUCTIVE REASONING: • what is the probability that those conclusions (or hypotheses) are true? P1: If it rains, the game is cancelled P2: the game is cancelled C: ? it rained

2. SOLVING PROBLEMS OF “LOGICAL FORM” • LOGIC is a formal system of rules of inference (algorithms) for evaluating the validity of arguments that draw conclusions from premises • REASONING is the human ability to evaluate such arguments • TWO TYPES OF LOGIC PROBLEMS: CONDITIONALCATEGORICAL PREMISE 1 if P, then Q All A are B PREMISE 2 P is true Some B are C CONCLUSION? Q is true ? Some A are C

3. THE CARD SELECTION TASK(Wason & Johnson-Laird, 1977) A K 4 7 Which card(s) need to be turned over to decide if the following rule is true: “if a card has a vowel on one side, then it has an even number on the other” ? Less than 5% of college students choose the correct cards. Why?

4. REASONING ABOUT CONDITIONAL PROBLEMSRips & Marcus, 1977 Premise 1: if P then Q (e.g., if the chair is green, the light is on) Premise 2OperationConclusion? %Corr P is trueaffirming theQ is true100% antecedent (modus ponens) P is falsedenying the ------- 79% antecedent Q is trueaffirming the ------- 77% consequent Q is falsedenying theP is false57% consequent (modus tolens) A K 4 7

5. SOURCES OF ERRORS IN CONDITIONAL REASONING • ENCODING • misinterpret the rule as “biconditional”Q if and only if P • fail to use appropriate schema“if beer is done, then 21”(Griggs & Cox, 1982) • SEARCH • fail to look for disconfirming cases (“confirmation bias”)

6. IMPROVING PERFORMANCE IN THE CARD SELECTION TASKPlatt, 1992 • (1) Clarify rule as conditional, not biconditional • (2) Require subjects to justify choices • (3) define task as a search for violations

7. CATEGORICAL SYLLOGISMS major premiseSome B’s are not A minor premiseNo C’s are B conclusion ? Some A’s are not C C A B argument is invalid!Conclusions must be true for all possible encodings and combinations of premises All men are mortal Socrates is a man ? All men are Socrates (W. Allen, 1975)

8. POCKET GUIDE FOR SOLVING CATEGORICAL PROBLEMS to reject show that premises as invalid: can be combined so: All A are B Some A are not B No A are B Some A are B Some A are B No A are B Some A are not B All A are B and, since most syllogisms are invalid, when in doubt, throw it out

9. SOURCES OF ERRORS IN CATEGORICAL REASONING • fail to make a valid inference:some B’s are A some A’s are Bno C’s are Bno B’s are C? some A’s are not C ? some A’s are not C 60% corr 80% corr • make an invalid inference (illicit conversion):all A’s are Ball C’s are B all B’s are C? all A’s are C • fail to systematically search problem space:no A’s are B all B’s are C? no A’s are C B A B C B C B C C B C B A A A

10. BELIEF BIAS IN DEDUCTIVE REASONING all A’s are B some B’s are c ? some A’s are C All sharks are animals some animals are pets ? some sharks are pets all dogs are animals some animals are mean ? some dogs are mean all women are bad drivers all wealthy people are republicans all professors are absent minded etc etc