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:
P2:
If it rains, the game is cancelled
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
P is true
A
P is false
K
Q is true
4
Q is false
7
Operation
affirming the
antecedent
Conclusion? %Corr
Q is true
100%
(modus ponens)
denying the
antecedent
-------
79%
affirming the
consequent
-------
77%
denying the
consequent
P is false
(modus tolens)
57%
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
p e rc e n t c o rre c t
• (1) Clarify rule as conditional, not
biconditional
• (2) Require subjects to justify
choices
• (3) define task as a search for
violations
100
80
60
40
20
0
0
1
1&2
instructions
1,2&3
CATEGORICAL SYLLOGISMS
major premise
minor premise
Some B’s are not A
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 mortal
Socrates is a man
? All men are Socrates
(W. Allen, 1975)
POCKET GUIDE FOR SOLVING
CATEGORICAL PROBLEMS
to reject
as invalid:
show that premises
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
SOURCES OF ERRORS IN
CATEGORICAL REASONING
• fail to make a valid inference:
some B’s are A
no C’s are B
? some A’s are not C
60% corr
some A’s are B
no B’s are C
? some A’s are not C
80% corr
• make an invalid inference (illicit
conversion):
all A’s are B
all 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
A
B
all B’s are C
B
C
B C
? no A’s are C
B C
A
B
C
A
B C
A
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