copyright © 2011 pearson education, inc. thinking critically 1c discussion paragraph 1 web 88....

Copyright © 2011 Pearson Education, Inc.

Thinking Critically

1C Discussion Paragraph

1 web88. State Politics89. US Presidents

90. Web Venn Diagrams

1 world91. Categorical Propositions92. Venn Diagrams in Your Life93. Quantitative Diagram

Copyright © 2011 Pearson Education, Inc. Slide 1-3

Unit 1D

Analyzing Arguments

1-D


Two Types of Arguments

Inductive Reasoning

specific premises → general conclusion

Example:

Premise: Bluebirds fly.Premise: Hummingbirds fly.Premise: Cardinals fly.Conclusion: All birds fly.

1-DHit MovieCN 1

A movie director tells her producer (who pays for the movie) not to worry—her film will be a hit. As evidence, she cites the following facts: She’s hired big stars for the lead roles, she has a great advertising campaign planned, and it’s a sequel to her last hit movie.

1. Explain why this argument is inductive, and evaluate its strength.


1-DEarthquakeCN 2

2. Evaluate the following argument and discuss the truth of its conclusion.

Geological evidence shows that, for thousands of years past, the San Andreas Fault has suffered a major earthquake at least once every hundred years. Therefore, we should expect another earthquake on the fault during the next one hundred years.


1-D


Two Types of Arguments

Deductive Reasoning:

general premises → specific conclusion

Example:

Premise: All doctors are intelligent.Premise: Dr. Jones is a doctor.Conclusion: Dr. Jones is intelligent.

1-D


Evaluating Arguments

An argument is strong if a compelling case is made for its conclusion.

An argument is weak if the conclusion is not well supported by its premises.

An inductive argument does not prove its conclusion true, so it is evaluated based on its strength.

1-D


Evaluating Arguments

The argument is valid if its conclusion follows necessarily from its premises, regardless of the truth of the premises or conclusion.

The argument is sound if it is valid and its premises are all true.

Apply two criteria to evaluate a deductive argument.

1-D


1. Draw a Venn diagram that represents all the information contained in the premises.

2. If the Venn diagram contains the conclusion the argument is valid; otherwise, it is not.

The following tests the validity of a deductive argument with a Venn diagram:

A Venn Diagram Test of Validity

1-D


All politicians are married.

Senator Harris is a politician.

Therefore, Senator Harris is married.


1-DInvalid ArgumentCN 3

Evaluate the validity and soundness of the following argument.

Premise: All fish live in the water. Premise: Whales are not fish. Conclusion: Whales do not live in the water.


1-D


All fish live in the water.

Whales are not fish.

Therefore, whales do not live in the water.


1-DInvalid but True ConclusionCN 4

Evaluate the validity and soundness of the following argument. Draw a Venn diagram to represent.

Premise: All 20th-century US presidents were men.

Premise: John Kennedy was a man. Conclusion: John Kennedy was a 20th century US

president.


1-D


Basic Forms of Conditional Deductive Arguments

1-DAffirming the Hypothesis (Valid)CN 5

5. Use a Venn diagram test to show that the argument is valid:

Premise: If a person lives in Chicago, then the person likes windy days.

Premise: Carlos lives in Chicago. Conclusion: Carlos likes windy days.


1-DAffirming the Conclusion (Invalid)CN 6

6. Use a Venn diagram to test the validity of the following argument.

Premise: If an employee is regularly late, then the employee will be fired.

Premise: Sharon was fired. Conclusion: Sharon was regularly late.


1-DDenying the Hypothesis (Invalid)CN 7

7. Use a Venn diagram to test the validity of the following argument:

Premise: If you liked the book, then you’ll love the movie.

Premise. You did not like the book. Conclusion: You will not love the movie.


1-DDenying the Conclusion (Valid)CN 8

8. Analyze the following argument:

Premise: A narcotic is habit-forming. Premise: Heroin is not habit forming. Conclusion: Heroin is not a narcotic.


1-D

Notes

Turn notes over to back side. 4 sections. Label Each section . We will draw a Venn Diagram containing arguments for each of the deductive

1. Affirm Hypothesis (p) Valid 2. Affirm Conclusion (q) Invalid 3. Deny Hypothesis (p) Invalid 4. Deny Conclusion (q) Valid


1-D


Affirming the Hypothesis (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. Valid(modus ponens)


1-D


Affirming the Conclusion (consequent):


Marilyn gets a good job.

Marilyn has a college degree.Invalid(inverse fallacy)


1-D


Denying the Hypothesis (antecedent):


Marilyn does not have a college degree.

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


1-D


Denying the Conclusion (consequent):


Marilyn does not have a good job.

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


1-DA Chain of ConditionalsCN 9

9. Analyze this argument.

“If elected to the school board, Maria Lopez will force the school district to raise academic standards, which will benefit my children’s education. Therefore, my children will benefit if Maria Lopez is elected.


1-DInvalid Chain of ConditionalsCN 10

10. Analyze the following argument: “We agreed that if you shop, I make dinner. We

also agreed that if you take out the trash, I make dinner. Therefore, if you shop you should take out the trash”


1-D

Examples (Deductive Arguments)

1. Rephrase if p then q 2. Identify type with a Venn Diagram 3. Truths 4. Is it sound?


1-D

Examples (Analyze Deductive)

Premise: All European countries use the euro as currency. Great Britain is a European Country.

Conclusion: Great Britain uses the euro as currency.

Premise. All dairy products contain protein. Soybeans contain protein.

Conclusion: Soybeans are dairy products.


1-DExamples con’t (Analyze Deductive)

Premise: No states west of the Mississippi River are in the eastern time zone. Utah is west of the Mississippi River.

Conclusion: Utah is not in the eastern time zone

Premise: If an animal is a dog, then it is a mammal. Setters are dogs.

Conclusion: Setters are mammals.


1-D

Notes (chain of conditionals) Turn notes over, label the 4th section Chain of Conditionals

Premise: If p then q, if q then r Conclusion: If p then r Valid

Premise: if p then q, if r then q Conclusion: if p then r InvalidCopyright © 2011 Pearson Education, Inc. Slide 1-30

1-D

Examples (chains)

Premise: If a natural number is divisible by 8, then it is divisible by 4. If a natural number is divisible by 4 , then it is divisible by 2.

Conclusion: If a natural number is divisible by 8 then it is divisible by 2


1-D

Proofs and Rules

A proof is a deductive argument. Claim is a Theorem in a proof.

Testing inductively a Rule Try a few examples If they are all true, then rule is valid If one is false, the rule is invalid


1-DInductively Testing a Mathematical RuleCN 11

11. Test the following rule:

For all numbers a and b, a x b = b x a


1-D


Inductive Counterexample

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

Based on the test cases to the right, it appears that n2 n 11 will always equal a prime number when n ≥ 0.

Or does it?

How about n = 11?

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

1-DInvalidating a Proposed RuleCN 12

Suppose you cannot recall whether adding the same amount to both the numerator and the denominator of a fraction such as 2/3 is legitimate.

12. That is you are wondering whether it is true that, for any number a

2 = 2+a/3+a


1-DHomework 1D Discussion Paragraph on 1C Class Notes 1-12 Quick Quiz p.61:1-10 Exercises p.62:1-14 1 Web

66. Fermat’s Last Theorem 67. Pythagorean Theorem

1 World 68. Deductive Reasoning in your Life 69. Inductive Reasoning in your Life 70. Editorial Arguments 71. Arguing Your Side 72. Arguing the Other Side


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