Copyright © 2005 Pearson Education, Inc. Slide 1-1.

Copyright © 2005 Pearson Education, Inc. Slide 1-1

Copyright © 2005 Pearson Education, Inc.

Chapter 1


Quantitative Reasoning

College Course Work

Career Experiences

Daily Life

Quantitative

Skills

Prologue


Six Math Misconceptions

1. Math requires a special brain.

2. Math in modern issues is too complex.

3. Math makes you less sensitive.

4. Math makes no allowance for creativity.

5. Math provides exact answers.

6. Math is irrelevant to my life.

Prologue


What is mathematics?

Sum of its branches Way to model the world

Language

Math

Prologue


Importance of Logic

Parade MagazineAsk Marilyn Column

Question: “What is the most important thing a person can do to improve his or her critical thinking skills?”

Phyllis Evitch, Rice Lake, Wisconsin Answer: “Study logic. Without a sound foundation in the principles

of reasoning, you’ll be less able to understand your world, and the ramifications of this will ripple through everything from work to play. Even worse, you won’t realize what you’re missing.”

Marilyn Vos Savant

1-A


Definitions

Logic is the study of the methods and principles

of reasoning. An argument uses a set of facts or assumptions,

called premises, to support a conclusion. A fallacy is a deceptive argument—an argument

in which the conclusion is not well supported

by the premises.

1-A


Fallacy Structures

Appeal to Popularity Many people believe p is true; therefore ... p is true.

False Cause A came before B; therefore ... A caused B.

Appeal to Ignorance There is no proof that p is true;

therefore ... p is false.

Hasty Generalization A and B are linked one or a few times;

therefore ... A causes B or vice versa.

Limited Choice p is false; therefore ... only q can be true.

1-A


Fallacy Structures

Appeal to Emotion p is associated with a positive emotional response; therefore . . . p is true.

Personal Attack I have a problem with the person or group claiming p. p is not true.

Circular Reasoning p is true. p is restated in different words.

Diversion (Red Herring) p is related to q and I have an argument concerning q; therefore . . . p is true.

Straw Man I have an argument concerning a distorted version of p; therefore . . . I hope you are fooled into concluding I have an argument concerning the real version of p.

1-A


Negations (Opposites)

1-B

p not p

T F F T


And Statements (Conjunctions)

p q p and q

T T T T F F F T F F F F

Note: Conjunction is false unless both p and q are true.

1-B


Or Statements (Disjunctions)

p q p or q

T T T T F T F T T F F F

Note: Disjunction is true unless both p and q are false.

1-B


If . . . Then Statements(Conditionals)

p q if p, then q

T T T T F F F T T F F T

Note: Conditional is true unless p is true and q is false.

1-B


Truth Table Practice

Note: ~ signifies NEGATION

signifies AND signifies OR

Practice by writing the truth values of each row in the table above.

1-B

p q ~ p ~ q p q ~ p ~ q ~ ( p q)

T T

T F

F T

F F



p q ~ p ~ q p q ~ p ~ q ~ ( p q)

T T F F T F F

T F

F T

F F


1-B





p q ~ p ~ q p q ~ p ~ q ~ ( p q)

T T F F T F F

T F F T F T T

F T

F F


1-B





p q ~ p ~ q p q ~ p ~ q ~ ( p q)

T T F F T F F

T F F T F T T

F T T F F T T

F F


1-B





p q ~ p ~ q p q ~ p ~ q ~ ( p q)

T T F F T F F

T F F T F T T

F T T F F T T

F F T T F T T


1-B




Converse, Inverse, and Contrapositive

Conditional: If it is raining, then I will bring an umbrella to work.

Converse: If I bring an umbrella to work, then it must be raining.

Inverse: If it is not raining, then I will not bring an umbrella to

work.

Contrapositive: If I do not bring an umbrella to work, then it must not be raining.

1-B


Definitions

A set is a collection of objects; the individual

objects are the members of the set. We often

describe sets by listing their members within a

pair of braces, {}. If there are too many members

to list, we can use three dots, …, to indicate a

continuing pattern.

1-C


Real Number Venn Diagram

1-C


Venn Diagram for Categorical Propositions

1-C

All S are P



1-C

No S are P



1-C

Some S are P



1-C

Some S are not P


Negations for Categorical Propositions

1-C

Proposition Negation

All S are P

No S are P

Some S are P

Some S are not P

Some S are not P

Some S are P

No S are P

All S are P


Venn Diagram of Blood Types

1-C


Two Types of Arguments

1-D

Inductive Reasoning

specific premises

general conclusion

Deductive Reasoning

general premises

specific conclusion


Basic Forms of Conditional Deductive Arguments

1-D

Valid(modus ponens)

Affirming the Antecedent:

If one gets a college degree, then one can get a good job.

Marilyn has a college degree.

Marilyn can get a good job.



1-D

Affirming the Consequent:


Marilyn gets a good job.

Marilyn has a college degree.Invalid(inverse fallacy)



1-D

Denying the Antecedent:


Marilyn does not have a college degree.

Marilyn cannot get a good job.Invalid(converse fallacy)



1-D

Denying the Consequent:


Marilyn does not have a good job.

Marilyn does not a college degree.Valid(modus tollens)


Inductive Counterexample

1-D

42 4 11 23 (prime)4

52 5 11 31 (prime)5

32 3 11 17 (prime)3

22 2 11 13 (prime)2

12 1 11 11 (prime)1

02 0 11 11 (prime)0

n2 n 11n

Consider the following algebraic expression: n2 n 11

It appears that n2 n 11will always equal a prime number when n ≥ 0.

Or does it?

How about n = 11?

112 11 + 11 = 121(a non-prime counterexample)


Critical Thinking In Everyday Life

General Guidelines.1. Read (or listen) carefully.2. Look for hidden assumptions.3. Identify the real issue.4. Use visual aids.5. Understand all the options.6. Watch for fine print and missing information.7. Are other conclusions possible?

1-E

Copyright © 2005 Pearson Education, Inc. Slide 1-1.

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