REASONING AS PROBLEM SOLVING

• DEDUCTIVE REASONING:– what, if any, conclusions

necessarily follow?

• INDUCTIVE REASONING:– what is the probability that those

conclusions (or hypotheses) are true?

given a set of facts (premises),

P1: If it rains, the game is cancelledP2: the game is cancelled

C: ? it rained

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:

CONDITIONAL CATEGORICAL

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

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?

REASONING ABOUT CONDITIONAL PROBLEMS

Rips & Marcus, 1977

Premise 1: if P then Q(e.g., if the chair is green, the light is on)

Premise 2 Operation Conclusion? %Corr

P is true affirming the Q is true 100%antecedent (modus ponens)

P is false denying the ------- 79%antecedent

Q is true affirming the ------- 77%consequent

Q is false denying the P is false 57%consequent (modus tolens)

A

K

4

7

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”)

IMPROVING PERFORMANCE IN THE CARD SELECTION TASK

Platt, 1992

• (1) Clarify rule as conditional, not biconditional

• (2) Require subjects to justify choices

• (3) define task as a search for violations

0

20

40

60

80

100

pe

rce

nt

co

rre

ct

0 1 1&2 1,2&3instructions

CATEGORICAL SYLLOGISMS

major premise Some B’s are not Aminor premise No 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 mortalSocrates is a man? All men are Socrates

(W. Allen, 1975)

POCKET GUIDE FOR SOLVING CATEGORICAL PROBLEMS

to reject show that premisesas 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

A

• fail to make a valid inference: some B’s are A some A’s are B no C’s are B no 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

AAA B

B

B

B

B

B

CCC

CC

BELIEF BIAS IN DEDUCTIVE REASONING

all A’s are Bsome B’s are c? some A’s are C

All sharks are animalssome animals are pets? some sharks are pets

all dogs are animalssome animals are mean? some dogs are mean

all women are bad driversall wealthy people are republicansall professors are absent minded

etc etc