Logical Reasoning zDeductive reasoning zInductive reasoning.

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