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.
1
LOGIC is a formal system of rules of
inference (algorithms) for evaluating the
validity of arguments that draw
conclusions from premises
Argument: 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
2
Two Types of Logic Problems
LOGIC 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
Categorical Problems
• 1
• 2
A&B
C
A
B
C
4
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
P is true affirming the
antecedent
A
P is false
Conclusion?
%Corr
Q is true
100%
(modus ponens)
denying the
antecedent
-------
79%
affirming the
consequent
-------
77%
K
Q is true
4
Q is false
7
denying the
P is false
consequent (modus tolens)
57%
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)
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 Some B’s are not A
minor premise No C’s are B
B
Conclusion ? Some A’s are not C
C
A
B A
C
B A C
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 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
To reject Conclusion:
All C are A
– Find the quantifier for C
• ALL C are A
– Find the negation for A
• ALL C 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
1 C AB
All students are take classes
Some football players do not take classes
Some football players are not students
2 C
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
• Some C 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
C B
A
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
• ALL C 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
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
• ALL C 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=B=C
A
B
C
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:
A B
no A’s are B
B
all B’s are C
B C
C
? no A’s are C
B C
A
B C
A
B
C
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
•
•
1)
B
•
A
1)
A
C
•
2)
•
B
A
C
C
B
•
3)
A
C
C
2)
B
A
B
3)
C
A
B
all A’s are B
all C’s are B
? all A’s are C
•
all B’s are C
? all A’s are C
•
1)
2)
C
C
B
A
B
A
BELIEF BIAS IN
DEDUCTIVE REASONING
all A’s are B
some B’s are c
? some A’s are C
A B
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