Logical Reasoning

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Logical Reasoning

• Deductive reasoning
• Inductive reasoning

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Deductive Reasoning

• Reasoning from the general to the specific
• For 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.

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Common deductive reasoning problems

• Series problems
• Syllogisms

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Series problems

• review series of statements
• arrive at a conclusion not contained in any
  single statement
• For example
• Robin is funnier than Billy
• Billy is funnier than Sinbad
• Whoopi is funnier than Billy
• Q Is Whoopi funnier than Sinbad

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Syllogisms

• Present two general premises that must be
  combined to see if a particular conclusion is true

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Syllogism Example

• All Intro to Psychology students love their
  instructor.
• You are all Intro to Psychology students.
• Must you love your instructor?

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Syllogism Example

• All chefs are violinists.
• Mary is a chef.
• Is Mary a violinist?

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Ways to solve syllogisms

• Mental model theories
• Pragmatic reasoning theories

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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?

YES!
YES!
YES!
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Mental models theories

• All Intro to Psychology students love their
  instructor.

• You are all Biology students.

• Must you love your instructor?

NO!
NO!
NO!
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Mental models theories

• Syllogisms that are easy to visualize are more
  readily solved than more abstract syllogisms

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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

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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

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Inductive reasoning

• Reasoning from the specific to the general

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Inductive reasoning

• 18 16 14 ?? ??

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• Rule? Decrease by 2
• Q Why inductive reasoning?
• Answer Take SPECIFIC numbers (i.e. 18,16,14) and
  come up with a GENERAL rule (i.e. decrease by 2)

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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

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Inductive reasoning problems

• 7 8 16 17 ?? ??

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• 4 8 5 10 ?? ?? ??

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11
14

• 720 120 24 ?? ?? ??

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2
1
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Inductive reasoning problems

• 5 10 15 ?? ?? ?? ?? ?? ??
  ?? ??

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55
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• 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?

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Confirmation bias

• Only search for information confirming ones
  hypothesis
• Example reading newspaper columnists who agree
  with our point of view and avoiding those who
  dont

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Chris story

• Chris is 67, 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?

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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?

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Representativeness

• Judge probability of an event based on how it
  matches a prototype
• Can be good
• But can also lead to errors
• Most will overuse representativeness
• i.e. Steves description fits our vision of a
  librarian

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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

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Guess the probabilities

• How many people die each year from
• Heart disease?
• Floods?
• Plane crashes?
• Asthma?
• Tornados?

Stop
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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 dont make the local news

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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?

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Class demonstration

• Name words starting with k
• Name words with the letter k in the third
  position

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Availability heuristic

• Because it is easier to recall words starting
  with k , people overestimate the number of
  words starting with k

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Finish the sequence problems

• 30 24 18 ?? ?? ??

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6
0

• Rule?
• Decrease by six

• 1 3 2 4 ?? ?? ?? ??

6
4
5
3

• Rule?
• Increase by two, decrease by 1

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Finish the sequence problems

• 2 3 10 12 ?? ?? ?? ?? ?? ?? ?? ?? ?? ??
  ?? ?? ?? ?? ?? ?? ?? ?? ?? ??
  ??

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200
299
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201
302
2000
399

• Rule?
• Increasing numbers starting with the letter t

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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 didnt play against each other.

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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 wasnt a
  bartender. He was a robber.

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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.

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Nine dots problem

• Without lifting your pencil or re-tracing any
  line, draw four straight lines that connect all
  nine dots

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Answer to nine dots problem
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Metal Set

• Q Why couldnt you solve the previous problems?
• A Mental set - a well-established habit of
  perception or thought

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Strategies for solving problems

• 1. Break mental sets

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Number problem mental set

• 2 3 10 12 ?? ?? ?? ?? ?? ?? ?? ?? ?? ??
  ?? ?? ?? ?? ?? ?? ?? ?? ?? ??
  ??

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200
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201
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2000
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• Most people get stuck in the same rhythm
• Only view problems in terms of math formulas
• Need to break out of this mental set to solve the
  problem

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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

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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

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Answer to candle problem

• Most people do not think of using the box for
  anything other than its normal use (to hold the
  tacks)
• To solve the problem, you have to overcome
  functional fixedness

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Functional fixedness

• type of mental set
• inability to see an object as having a function
  other than its usual one

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Strategies for solving problems

• 1. Break mental sets
• break functional fixedness
• 2. Find useful analogy

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Find useful analogy

• Compare unknown problem to a situation you are
  more familiar with

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Strategies for solving problems

• 1. Break mental sets
• 2. Find useful analogy
• 3. Represent information efficiently
• 4. Find shortcuts (use heuristics)

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Two general classes of rules for problem solving

• 1. Algorithms
• 2. Heuristics

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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

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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?

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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

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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.

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Strategies for solving problems

• 1. Break mental sets
• 2. Find useful analogy
• 3. Represent information efficiently
• 4. Find shortcuts
• 5. Establish subgoals
• 6. Turn ill-defined problems into well-defined
  problems