Thinking Critically

1. 1 Thinking Critically Analyzing Arguments

2. Two Types of Arguments Inductive: specific premises → general conclusion Example: Premise: Bluebirds fly. Premise: Hummingbirds fly. Premise: Cardinals fly. Conclusion: All birds fly.

3. Two Types of Arguments Deductive: general premises → specific conclusion Example: Premise: All doctors are intelligent. Premise: Dr. Jones is a doctor. Conclusion: Dr. Jones is intelligent.

4. Evaluating Inductive Arguments An inductive argument does not prove its conclusion true, so it is evaluated based on its strength. • 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.

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

6. Example-cont Solution Each of the three pieces of evidence is a specific characteristic of her movie. She uses them to support the more general conclusion that her movie will be a hit. Because the conclusion is more general than the premises, the argument is inductive. In this case, her argument is relatively weak. As all producers know, even the best planned movies can flop.

7. Evaluating Arguments Apply two criteria to evaluate a deductive argument. • 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.

8. Key Distinctions: Inductive and Deductive Arguments

9. A Venn Diagram Test of Validity The following tests the validity of a deductive argument with a Venn diagram: 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.

10. A Venn Diagram Test of Validity All fish live in the water. Whales are not fish. Whales do not live in the water.

11. Basic Forms of Conditional Deductive Arguments

12. Example Consider the following argument: Premise: If a person lives in Chicago, then the person likes windy days. Premise: Carlos lives in Chicago. Conclusion: Carlos likes windy days.

13. Example (cont) Use a Venn diagram test to show that the argument concerning Carlos and Chicago is valid. The first premise is if p, then q, in which p = a person lives in Chicago and q = the person likes windy days. We therefore draw a Venn diagram with the p circle inside the q circle.

14. Example (cont) Use a Venn diagram test to show that the argument concerning Carlos and Chicago is valid. The second premise asserts that p is true for Carlos. We therefore draw an X to represent Carlos and place it inside the p circle.

15. Example (cont) Use a Venn diagram test to show that the argument concerning Carlos and Chicago is valid. The conclusion states that q is true for Carlos (he likes windy days), which means that if the argument is valid, we expect to see the X that represents Carlos inside the q circle. The X is indeed within the q circle, so the argument is valid.

16. Example 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. Solution Start with the Venn diagram for if p, then q, which means the p circle inside the q circle In this case, p = an employee is regularly late and q = the employee will be fired.

17. Example (cont) The second premise asserts that q is true for the person named Sharon (she was fired). We draw an X to represent Sharon and place it inside the q circle. However, because the premise does not tell us whether Sharon was regularly late, we place the X on the border of the p circle, indicating that we don’t know whether the X belongs inside or outside this circle.

18. Example (cont) The conclusion states that p is true for Sharon (she was regularly late). Therefore, if the conclusion is supported by the premises, the diagram should have the X that represents Sharon inside the p circle. But it doesn’t—the X is on the border, so the argument is invalid.

19. Example 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. Solution Start with the Venn diagram for if p, then q, with p = you liked the book and q = you’ll love the movie.

20. Example (cont) The second premise asserts that p is false for you, so we put an X (to represent you) outside the p circle. Because we do not know whether the X should also be outside the q circle (the premise does not tell us whether you’ll love the movie), we place it on the border of the q circle. The conclusion states that q is false for you (you will not love the movie), so if the argument is valid, we expect the X to be outside the q circle. It is not, so the argument is invalid. In other words, it’s possible that you will still love the movie even though you did not like the book.

21. Example (cont)

22. Example Use a Venn diagram to test the validity of the following argument. Premise: A narcotic is habit-forming. Premise: Aspirin is not habit-forming. Conclusion: Aspirin is not a narcotic. Solution We must start by rephrasing the first premise in standard conditional form by writing if a substance is a narcotic, then it is habit-forming. We identify p = a substance is a narcotic and q = the substance is habit-forming and draw the Venn diagram.

23. Example The second premise asserts that q is false for aspirin (it is not habit-forming). We therefore place an X, representing aspirin, outside the q circle. The conclusion states that p is false for aspirin (it is not a narcotic), which means the X representing aspirin should be outside the narcotic circle—and it is, so the argument is valid.

24. Example

25. Deductive Arguments with a Chain of Conditionals Another common type of deductive argument involves a chain of three or more conditionals. Such arguments have the following form: • Premise: If p, then q. • Premise: If q, then r. • Conclusion: If p, then r. This particular chain of conditionals is valid: If p implies q and q implies r, it must be true that p implies r.

26. Example Determine the validity of 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.” Solution This argument can be rephrased as a chain of conditionals: • Premise: If Maria Lopez is elected to the school board, then the school district will raise academic standards.

27. Example • Premise: If the school district raises academic standards, then my children will benefit. • Conclusion: If Maria Lopez is elected to the school board, then my children will benefit. Cast in this form, the conditional propositions form a clear chain from p = Maria Lopez is elected to q = the district will raise academic standards to r = my children will benefit. Therefore, the argument is valid.

28. Induction and Deduction Perhaps more than any other subject, mathematics relies on the idea of proof. A mathematical proof is a deductive argument that demonstrates the truth of a certain claim, or theorem. A theorem is considered proven if it is supported by a valid and sound proof. Although mathematical proofs use deduction, theorems are often discovered by induction.

29. Example Test the following rule: For all numbers a and b, a × b = b × a. Solution We begin with some test cases, using a calculator as needed. Does 7 × 6 = 6 × 7? Yes! Does (–23.8) × 9.2 = 9.2 × (–23.8)? Yes! Does Yes!

30. Example (cont) The three test cases are each somewhat different (mixing fractions, decimals, and negative numbers), yet the rule works in all three cases. This outcome offers a strong inductive argument in favor of the rule. Although we have not proved the rule a × b = b × a, we have good reason to believe that it is true. Our belief would be strengthened by additional test cases that confirm the rule.