Copyright © 2011 Pearson Education, Inc. Thinking Critically

Copyright © 2011 Pearson Education, Inc. Slide 1-3 Unit 1D Analyzing Arguments

Notes Get out a piece of paper, fold into 4 sections. Section 1 First section, draw line in half. Label each half: Inductive Reasoning Deductive Reasoning Copyright © 2011 Pearson Education, Inc. Slide 1-4

1-D Copyright © 2011 Pearson Education, Inc. Slide 1-5 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-D Copyright © 2011 Pearson Education, Inc. Slide 1-6 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 Examples (Inductive or Deductive) Hit Movie (p.52) I have never found mail in my mailbox on a Sunday. The Postal Service must not have Sunday deliveries. Because of a budget cutback, postal workers will no longer work on Saturdays. Therefore, I will not expect Saturday deliveries on time. Copyright © 2011 Pearson Education, Inc. Slide 1-7

1-D Example (con’t Inductive or Deductive) Eminem’s first five CDs were outstanding. His next CD is bound to be good so I will buy it without even listening to it. If I eat spicy food before noon, then I get indigestion, I have no appetite for the next six hours. Therefore, if I eat spicy food before noon then I cannot eat dinner. January is windier than July. The wind must blow more often in the winter than in the summer. Copyright © 2011 Pearson Education, Inc. Slide 1-8

1-D Examples (Analyze Inductive) Earthquake (p.52) True/False Weak/Strong Premise: 2+3=5, 5+4=9, 7+6=13 Conclusion: The sum of an even integer and an odd integer is an odd integer. Premise: If I pay more for a pair of running shoes, they last longer. If I pay more for an automobile, it requires fewer repairs. Conclusion: Quality goes with higher prices. Copyright © 2011 Pearson Education, Inc. Slide 1-9

1-D Examples (Analyze Inductive) Premise: Trout and bass swim and they are fish. Sharks and marlin swim and they are fish. Tuna and salmon swim and they are fish. Conclusion: Whales swim and they are fish. Premise: Apes and baboons have hair and they are mammals. Mice and rats have hair and they are mammals. Tigers and lions have hair and they are mammals. Conclusion: animals with hair are mammals. Copyright © 2011 Pearson Education, Inc. Slide 1-10

1-D Notes Section 2 Analyze Inductive Reasoning Premises (truths) Conclusion (truths) Strength of Argument Weak Strong Copyright © 2011 Pearson Education, Inc. Slide 1-11

1-D Copyright © 2011 Pearson Education, Inc. Slide 1-12 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 Notes Section 3 Analyze Deductive Arguments Rephrase - All S are P Draw Venn Diagram Premises are truth Sound Argument In evaluating deductive, we can be sure that the conclusion is true if answer to both questions is yes Copyright © 2011 Pearson Education, Inc. Slide 1-13

1-D Copyright © 2011 Pearson Education, Inc. Slide 1-14 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 Copyright © 2011 Pearson Education, Inc. Slide 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 Copyright © 2011 Pearson Education, Inc. Slide 1-16 All politicians are married. Senator Harris is a politician. Therefore, Senator Harris is married. A Venn Diagram Test of Validity

1-D Copyright © 2011 Pearson Education, Inc. Slide 1-17 All fish live in the water. Whales are not fish. Therefore, whales do not live in the water. A Venn Diagram Test of Validity

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 Copyright © 2011 Pearson Education, Inc. Slide 1-18

1-D Copyright © 2011 Pearson Education, Inc. Slide 1-19 Basic Forms of Conditional Deductive Arguments

1-D Copyright © 2011 Pearson Education, Inc. Slide 1-20 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) Basic Forms of Conditional Deductive Arguments

1-D Copyright © 2011 Pearson Education, Inc. Slide 1-21 Affirming the Conclusion (consequent): If one gets a college degree, then one can get a good job. Marilyn gets a good job. Marilyn has a college degree. Invalid (inverse fallacy) Basic Forms of Conditional Deductive Arguments

1-D Copyright © 2011 Pearson Education, Inc. Slide 1-22 Denying the Hypothesis (antecedent): If one gets a college degree, then one can get a good job. Marilyn does not have a college degree. Marilyn cannot get a good job. Invalid (converse fallacy) Basic Forms of Conditional Deductive Arguments

1-D Copyright © 2011 Pearson Education, Inc. Slide 1-23 Denying the Conclusion (consequent): If one gets a college degree, then one can get a good job. Marilyn does not have a good job. Marilyn does not a college degree. Valid (modus tollens) Basic Forms of Conditional Deductive Arguments

1-D Examples (Deductive Arguments) 1. Rephrase if p then q 2. Identify type with a Venn Diagram 3. Truths 4. Is it sound? Copyright © 2011 Pearson Education, Inc. Slide 1-24

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. Copyright © 2011 Pearson Education, Inc. Slide 1-25

1-D Examples 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. Copyright © 2011 Pearson Education, Inc. Slide 1-26

1-D Notes (chain of conditionals) Turn notes over, label the 4 th 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 Invalid Copyright © 2011 Pearson Education, Inc. Slide 1-27

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 Copyright © 2011 Pearson Education, Inc. Slide 1-28

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 Copyright © 2011 Pearson Education, Inc. Slide 1-29

1-D Copyright © 2011 Pearson Education, Inc. Slide 1-30 Inductive Counterexample 4 2  4  11  23 (prime)  5  11  31 (prime)  3  11  17 (prime)  2  11  13 (prime)  1  11  11 (prime)  0  11  11 (prime) 0 n 2  n  11 n Consider the following algebraic expression: n 2  n  11 Based on the test cases to the right, it appears that n 2  n  11 will always equal a prime number when n ≥ 0. Or does it? How about n = 11? 11 2  = 121 (a non-prime counterexample)