Logical Reasoning
Deductive reasoningInductive reasoning
Deductive Reasoning
Reasoning from the general to the specificFor example, start with a general
statement: All cars have tires.You can apply this general statement to
specific instances and deduce that a Ford Escort, a Toyota Camry, and a Mercedes Benz must have tires.
Common deductive reasoning problems
Series problemsSyllogisms
Series problems
review series of statementsarrive at a conclusion not contained in any
single statementFor example:Robin is funnier than BillyBilly is funnier than SinbadWhoopi is funnier than BillyQ: Is Whoopi funnier than Sinbad
Syllogisms
Present two general premises that must be combined to see if a particular conclusion is true
Syllogism Example
All Intro to Psychology students love their instructor.
You are all Intro to Psychology students.
Must you love your instructor?
Syllogism Example
All chefs are violinists.Mary is a chef.Is Mary a violinist?
Ways to solve syllogisms
Mental model theoriesPragmatic reasoning theories
Mental models theories To solve a syllogism,
you might visualize the statements
All Intro to Psychology students love their instructor.
You are all Intro to Psychology students.
Must you love your instructor?
Psych-ology
Psych-ology
Psych-ology
Bi-ology
Bi-ology
Bi-ology
Bi-ology
Mental models theories
All Intro to Psychology students love their instructor.
You are all Biology students.
Must you love your instructor?
Psych-ology
Psych-ology
Psych-ology
Bi-ology
Bi-ology
Bi-ology
Bi-ology
Mental models theories
Syllogisms that are easy to visualize are more readily solved than more abstract syllogisms
Psych-ology
Psych-ology
Psych-ology
Bi-ology
Bi-ology
Bi-ology
Bi-ology
Mental model theories
To solve a syllogism, you might visualize the statements
Syllogisms that are easy to visualize are more readily solved than more abstract syllogisms
Pragmatic reasoning theories
Solve syllogisms by applying information to pre-existing schemas
Problem difficulty related to importance of problem to our lives and survival as a species
More relevant = easier to solve
Inductive reasoning
Reasoning from the specific to the general
Inductive reasoning
18 16 14 ?? ??12 10
Rule? Decrease by 2Q: Why inductive reasoning? Answer: Take SPECIFIC numbers (i.e.
18,16,14) and come up with a GENERAL rule (i.e. decrease by 2)
Inductive Reasoning
Sherlock Holmes is perhaps a better example of INDUCTIVE reasoning than deductive reasoning
He takes specific clues and comes up with a general theory
Inductive reasoning problems
7 8 16 17 ?? ??
4 8 5 10 ?? ?? ??
25 26
117 14
720 120 24 ?? ?? ??6 2 1
Inductive reasoning problems
5 10 15 ?? ?? ?? ?? ?? ?? ?? ??2520 30 40 45 50 5535
Rule? Increase by five
WRONG!!!!! What is the correct rule? Any increasing number
- the next number could be 87 or 62 or 1,000,006
Why did everyone guess the wrong rule?
Confirmation bias
Only search for information confirming one’s hypothesis
Example: reading newspaper columnists who agree with our point of view and avoiding those who don’t
Chris is 6’7”, 300 pounds, has 12 tattoos, was a champion pro wrestler, owns nine pit bulls and has been arrested for beating a man with a chain.
Is Chris more likely to be a man or a woman?
A motorcycle gang member or a priest?How did you make your decision?
Chris story
Steve story
Steve is meek and tidy, has a passion for detail, is helpful to people, but has little real interest in people or real-world issues.
Is Steve more likely to be a librarian or a salesperson?
How did you come to your answer?
Representativeness
Judge probability of an event based on how it matches a prototype
Can be goodBut can also lead to errorsMost will overuse representativeness
i.e. Steve’s description fits our vision of a librarian
Most will underuse base rates
Base rate - probability that an event will occur or fall into a certain category Did you stop to consider that there are a lot
more salespeople in the world than librarians?
By sheer statistics, there is a greatly likelihood that Steve is a salesperson.
But very few take this into account
Guess the probabilities
How many people die each year from:
Heart disease?Floods?Plane crashes? Asthma?Tornados?
Stop
Availability heuristic
Judge probability of an event by how easy you can recall previous occurrences of that event.
Most will overestimate deaths from natural disasters because disasters are frequently on TV
Most will underestimate deaths from asthma because they don’t make the local news
Word probabilities
Is the letter “k” most likely to occur in the first position of a word or the third position?
Answer: “k” is 2-3 times more likely to be in the third position
Why does this occur?
Class demonstration
Name words starting with “k”Name words with the letter “k” in the
third position
Availability heuristic
Because it is easier to recall words starting with “k” , people overestimate the number of words starting with “k”
Finish the sequence problems
30 24 18 ?? ?? ??12 6 0
1 3 2 4 ?? ?? ?? ??
Rule?Decrease by six
Rule?Increase by two, decrease by 1
6453
Finish the sequence problems
2 3 10 12 ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ??
Rule?Increasing numbers starting with
the letter “t”
13 2131 39 200 201 299 300 301
20 29 3032
302 2000 399
22
Chess problem
Two grandmasters played five games of chess. Each won the same number of games and lost the same number of games. There were no draws in any of the games. How could this be so?
Solution: They didn’t play against each other.
Bar problem
A man walked into a bar and asked for a drink. The man behind the bar pulled out a gun and shot the man. Why should that be so?
Solution: The man behind the bar wasn’t a bartender. He was a robber.
Bar problem # 2
A man who wanted a drink walked into a bar. Before he could say a word he was knocked unconscious. Why?
Solution: He walked into an iron bar, not a drinking establishment.
Nine dots problem
Without lifting your pencil or re-tracing any line, draw four straight lines that connect all nine dots
Answer to nine dots problem
Metal Set
Q: Why couldn’t you solve the previous problems?
A: Mental set - a well-established habit of perception or thought
Strategies for solving problems
1. Break mental sets
Number problem mental set
Most people get stuck in the same rhythmOnly view problems in terms of math
formulasNeed to break out of this mental set to
solve the problem
2 3 10 12 ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ??
13 2131 39 200 201 299 300 301
20 29 3032
302 2000 399
22
Nine dots mental set
Most people will not draw lines that extend from the square formed by the nine dots
To solve the problem, you have to break your mental set
Mounting candle problem
Using only the objects present on the right, attach the candle to the bulletin board in such a way that the candle can be lit and will burn properly
Answer to candle problem
Most people do not think of using the box for anything other than it’s normal use (to hold the tacks)
To solve the problem, you have to overcome functional fixedness
Functional fixedness
type of mental setinability to see an object as
having a function other than its usual one
Strategies for solving problems
1. Break mental sets break functional fixedness
2. Find useful analogy
Find useful analogy
Compare unknown problem to a situation you are more familiar with
Strategies for solving problems
1. Break mental sets2. Find useful analogy3. Represent information efficiently4. Find shortcuts (use heuristics)
Two general classes of rules for problem solving
1. Algorithms2. Heuristics
Two general classes of rules for problem solving
Algorithms - things the vice-president might say
Algorithms - rules that, if followed correctly, will eventually solve the problem
An algorithm example
Problem: List all the words in the English language that start with the letter “q”
If using an algorithm, would have to go through every single possible letter combination and determine if it were a word i.e. is “qa” a word; is “qb” a word etc. This would take a very long time
Instead, what rule could you use to eliminate these steps?
Rules for “q” problem
Skip ahead and assume the second letter is a “u”
Assume the third letter has to be a vowel
These types of rules are called heuristics
Heuristics
Any rule that allows one to reduce the number of operations that are tried in problem solving
a.k.a rules of thumb or shortcuts Another common heuristic:
Problem: List all the numbers from 1-100,000 that are evenly divisible by 5
Answer: Rather than divide each and every number, you would use the rule: Any number ending in 0 or 5 is evenly divisible by 5.
1. Break mental sets2. Find useful analogy3. Represent information efficiently4. Find shortcuts5. Establish subgoals6. Turn ill-defined problems into well-
defined problems
Strategies for solving problems