CG42_00_12.1
Inductive Reasoning
Inductive reasoning: conclusions drawn are
probable, not certain
Examples:
Robins have hollow bones.
Sparrows have hollow bones.
Therefore, birds have hollow bones.
The cork is wood and it floats.
The cube is wood and it floats.
Therefore, wooden things float.
If a burglar is in the house, then the door
will be ajar.
The door is ajar.
Therefore, a burglar is in the house.
Characteristics of inductive reasoning
1. Inductive processes produce an increase in knowledge.
- if the conclusion is true, it reduces the set of alternative possibilities
2. Inductive processes are risky.
- the conclusion may or may not be true
- new knowledge is uncertain
3. Inductive processes need constraints.
- an infinite number of conclusions can be drawn from a set of premises
The cork is wood and it floats.
The cube is wood and it floats.
Therefore, wooden things float.
Therefore, all things float.
Therefore, brown things float.
Therefore, light things float.
etc.
- induction must depend upon knowledge of world; knowledge provides constraints
Reasoning in probabilistic situations
- how do people reason in probabilistic situations?
- how should people reason in these situations?
Bayes’s theorem
If a burglar is in the house, then the door
will be ajar.
The door is ajar.
Therefore, a burglar is in the house.
- What is the probability that the conclusion is true?
- can calculate this with Bayes’s theorem
- based on several probabilities
- prior probability: the probability that the conclusion is true regardless of the evidence
H = house has been burglarized
P(H) = 0.001
P(not H) = P(~H) = 0.999
- conditional probability: probability that a particular type of evidence is true if a conclusion is true
D = door is ajar
P(door is ajar, given burglar)
= P(D | H) = 0.8
P(door is ajar, if no burglar)
= P(D | ~H) = 0.01
- posterior probability: probability that the conclusion is true, given the evidence
- P(H | D)
- note: probability is low, even though the open door is good evidence for burglary, and not for normal state of affairs
- why so low?
- because prior probability is low
- Bayes’s theorem is completely accurate
- provides a prescriptive (normative) model
- need a descriptive model
Base Rate Neglect
- people often ignore prior probabilities (base rates)
- Kahneman & Tversky (1973)
- cond. 1: individual chosen from a group of
70 engineers and 30 lawyers
- cond. 2: individual chosen from a group of
70 lawyers and 30 engineers
- how likely is it that a randomly selected person is an engineer?
- cond 1: 0.70 cond 2: 0.30
- what about this person:
Jack is a 45-year-old man. He is married and has four children. He is generally conservative, careful, and ambitious. He shows no interest in political and social issues and spends most of his free time on his many hobbies, which include home carpentry, sailing, and mathematical puzzles.
- how likely is it that Jack is an engineer?
- cond 1: 0.90cond 2: 0.90
- Tversky & Kahneman (1974)
Steve is very shy and withdrawn, invariably helpful, but with little interest in people or the world of reality. A meek and tidy soul, he has a need for order and structure and a passion for detail.
- is it more likely that Steve is a librarian, or a farmer?
- Most people pick librarian, even though the prior probability of being a farmer is much higher
- Hammerton (1973)
“Suppose you are tested for a rare type of cancer. It is known that having this type of cancer will produce a positive test result 95% of the time. If a person does not have cancer, the test will be positive 5% of the time. Suppose your result is positive. How likely is it that you have cancer?”
- most people say ~95%
- but, consider base rates!
- rare form of cancer: 1 in 10,000
- from Bayes’s theorem: posterior probability is 0.0019; about 1 in 500
- Gigerenzer & Hoffrage (1995): base rate neglect is reduced when reasoning about frequencies rather than probabilities
- probability group: 20% correct
- frequency group: 50% correct
- we experience frequencies of events, not probabilities
Scientific Reasoning
- how do people construct and test hypotheses?
- Wason’s (1960) rule discovery task
- what rule is used to generate this sequence? 2, 4, 6
- possibilities:
- increasing by two
- consecutive even numbers
- increasing numbers
- generate hypothesis; test by constructing triplets for feedback
- confirmation bias: subjects tested triplets that were consistent with hypothesis
- e.g., hypothesis is “increasing by 2”
- test with:
- 4, 6, 8; 5, 7, 9; 13.5, 15.5, 17.5
- not with:
- 4, 5, 6; 10, 16, 22
- Wason: this is an ineffective strategy: scientific reasoning should involve disconfirmation, not confirmation
- Klayman & Ha (1987):
- positive test strategy: test cases that are known to have property of interest
- can be effective under many common conditions
- two ways to seek disconfirmation
- testing cases that do not fit hypothesis
- testing cases that are likely to falsify hypothesis
- These are not always the same cases!
- positive test strategy: general default heuristic
- used in the absence of other evidence