Reasoning as problem solving

1. Reasoning as problem solving • What is REASONING? Process of making inferences or coming to conclusions from given information. • Deductive Reasoning: • Given a set of facts, (premises) what, if any, conclusions necessarily follow? The metallic parts of a motorcycle are hot after use -> The exhaust pipe is hot after use. • Inductive reasoning • What is the probability that those conclusions (or hypotheses) are true? The exhaust pipe on a motorcycle is hot after use -> All metallic parts must be hot after use.

2. LOGIC is a formal system of rules of inference (algorithms) for evaluating the validity of arguments that draw conclusions from premisesArgument: Premise 1 Premise 2 Conclusion Two Types of Logic Problems: Conditional and Categorical • Terminology for conditionals • Antecedent: P • Consequent: Q • Affirming: Saying true • Denying: Saying false

3. Two Types of Logic ProblemsLOGIC is a formal system of rules of inference (algorithms) for evaluating the validity of arguments that draw conclusions from premises Conditional • Premise 1: • If P, then Q • Premise 2: • P is true • Conclusion: • Q is true? Categorical • Premise 1: • All A are B • Premise 2: • Some B are C • Conclusion: • Some A are C? 3

4. 1 2 Categorical Problems A & B B A C C

5. THE CARD SELECTION TASK(Wason & Johnson-Laird, 1977) K 4 7 A 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?

6. 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 the Q is true100% antecedent (modus ponens) P is falsedenying the ------- 79% antecedent Q is trueaffirming the ------- 77% consequent Q is falsedenying theP is false 57% consequent (modus tolens) A K 4 7

7. SOURCES OF ERRORS IN CONDITIONAL REASONING SEARCH fail to look for disconfirming cases (“confirmation bias”) 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)

8. 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

9. CATEGORICAL SYLLOGISMS B A major premiseSome B’s are not A minor premiseNo C’s are B Conclusion ? Some A’s are not C B C B A 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)

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

11. To reject Conclusion: All C are A • Find the quantifier for C • ALL C are A • Find the negation for A • ALLC are __ A • Switch the quantifier and negation. • SOME C are NOT A New Conclusion: Some C are not A. Major premise: All A are B. Minor Premise: Some C are not B. Conclusion: Some C are not A. OR All students are take classes Some football players do not take classes Some football players are not students B 1 A C 2 C

12. To reject Conclusion: No C are A • Rewrite to an equivalent statement: • ALL C are NOT A • Find the quantifier for C • ALL C are NOT A • Find the negation for A • ALL C are NOT A • Switch the quantifiers and negation • SomeC are __ A New Conclusion: Some C are A. Major premise: All B are A. Minor premise: Some B are C. Conclusion: Some C are A. OR ALL students go to class Some students are football players Some football players go to class A C B

13. To reject Conclusion: Some C are A • Find the quantifier for C • SOME C are A • Is A negated? (NO) • SOME C are ___ A • Switch the quantifiers and negate A • ALLC are NOT A • Rewrite to an equivalent statement: • NO C are A New Conclusion: No C are A. Major premise: No A are B. Minor premise: All C are B. Conclusion: No C are A. OR No birds are pigs All swine are pigs No swine are birds A C B

14. To reject Conclusion: Some C are not A • Find the quantifier for C • SOME C are not A • 2nd: Find the quantifier for A • SOME C are NOT A • Switch the quantifiers • ALLC are ___ A New Conclusion: All C are A. Major premise: All B are A. Minor Premise: All C are B. Conclusion: All C are A. OR Al birds are animals All eagles are birds All eagles are animals OR A A=B=C B C

15. 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 SOURCES OF ERRORS IN CATEGORICAL REASONING B A B C B C B C A B C A B A C

16. 1) 2) 3) 1) 2) 3) 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 B A A B C C B C A C A B B A C C A B

17. 1) 2) all A’s are B all C’s are B all B’s are C? all A’s are C ? all A’s are C C C B B A A

18. 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 B C A all women are bad drivers all wealthy people are republicans all professors are absent minded etc etc