Thinking Critically Discussion 1 web 1 world 41. Campaign Site 42. Gun Control Debate 43. Death penalty Debate 44. Health Care Debate 45. Fallacy Web Sites 1 world 46. Editorial Fallacies 47. Fallacies in Advertising 48. Fallacies in Politics 49. Personal Fallacies Copyright © 2011 Pearson Education, Inc.

Propositions and Truth Values Unit 1B Propositions and Truth Values Copyright © 2011 Pearson Education, Inc.

Definitions A proposition makes a claim (either an assertion or a denial) that may be either true or false. It must have the structure of a complete sentence. Any proposition has two possible truth values: T = true or F = false. A truth table is a table with a row for each possible set of truth values for the propositions being considered. Copyright © 2011 Pearson Education, Inc.

Negation (Opposites) The negation of a proposition p is another proposition that makes the opposite claim of p. ← If p is true (T), not p is false (F). ← If p is false (F), not p is true (T). Symbol: ~ Copyright © 2011 Pearson Education, Inc.

Negation CN 1a-b 1.Consider the proposition – Betsy is the fastest runner on the team. a. Write its negation. b. If the negation is false, is Betsy really the fastest runner on the team? Copyright © 2011 Pearson Education, Inc.

Double Negation The double negation of a proposition p, not not p, has the same truth value as p. Copyright © 2011 Pearson Education, Inc.

Radiation and Health CN 2 After reviewing data showing an association between low-level radiation and cancer among older workers at the Oak Ridge National Laboratory, a health scientist from the U of North Carolina was asked about the possibility of a similar association among younger workers at another national laboratory. He was quoted as saying: My opinion is that it’s unlikely that there is no association. 2. Does the scientist think there is an association between low-level radiation and cancer? Copyright © 2011 Pearson Education, Inc.

The 2000 Miranda Ruling CN 3 In a June 2000 decision, the US Supreme Court voted 7-2 to uphold the basic requirements of the 1966 Miranda decision. That decision required that suspects taken into custody be informed of their constitutional rights, such as the right to remain silent and the right to legal counsel. In his majority opinion, Chief Justice William Rehnquist wrote: …legal principles weigh heavily against overruling Miranda now. 3. According to this statement, did the Chief Justice feel that legal principles support or oppose the original Miranda decision? Copyright © 2011 Pearson Education, Inc.

Logical Connectors Propositions are often joined with logical connectors—words such as and, or, and if…then. Example: p = I won the game. q = It was fun. Logical Connector and or if…then New Proposition I won the game and it was fun. I won the game or it was fun. If I won the game, then it was fun. Copyright © 2011 Pearson Education, Inc.

And Statements (Conjunctions) Given two propositions p and q, the statement p and q is called their conjunction. It is true only if p and q are both true. Symbol: Copyright © 2011 Pearson Education, Inc.

And Statements CN 4 4. Evaluate the truth value of the following two statements: a. The capital of France is Paris and Antarctica is cold. b. The capital of France is Paris and the capital of America is Madrid. Copyright © 2011 Pearson Education, Inc.

Or Statements (Disjunctions) The word or can be interpreted in two distinct ways: An inclusive or means “either or both.” An exclusive or means “one or the other, but not both.” In logic, assume or is inclusive unless told otherwise. Copyright © 2011 Pearson Education, Inc.

Inclusive or Exclusive CN 6 Kevin’s insurance policy states that his house is insured for earthquakes, fire, or robbery. Imagine that a major earthquake levels much of his house, the rest burns in a fire, and his remaining valuables are looted in the aftermath. 6. Would Kevin prefer that the or in his insurance policy be inclusive or exclusive? Why? Copyright © 2011 Pearson Education, Inc.

Or Statements (Disjunctions) Given two propositions p and q, the statement p or q is called their disjunction. It is true unless p and q are both false. Symbol: Copyright © 2011 Pearson Education, Inc.

Smart Cows? CN 7 Consider the statement: Airplanes can fly or cows can read. 7. Is it true? Copyright © 2011 Pearson Education, Inc.

Key Word Search CN 8 You are doing a research report for which you need to find articles about the federal debt that appeared in either Time or Newsweek between 2008 and 2010. You have access to a database that allows you to search on date, magazine, and title. 8. How should you structure the search? Copyright © 2011 Pearson Education, Inc.

If… Then Statements (Conditionals) A statement of the form if p, then q is called a conditional proposition (or implication). It is true unless p is true and q is false. Proposition p is called the hypothesis. Proposition q is called the conclusion. Copyright © 2011 Pearson Education, Inc.

Conditional Truths CN 9 9. Evaluate the truth of the statement: If 2 + 2 = 5, then 3 + 3 = 4 Copyright © 2011 Pearson Education, Inc.

Truth Table Practice Note: ~ signifies NEGATION  signifies AND  signifies OR Practice by writing the truth values of each row in the table above. Copyright © 2011 Pearson Education, Inc.

Truth Table Practice Note: ~ signifies NEGATION  signifies AND  signifies OR Practice by writing the truth values of each row in the table above. Copyright © 2011 Pearson Education, Inc.

Truth Table Practice Note: ~ signifies NEGATION  signifies AND  signifies OR Practice by writing the truth values of each row in the table above. Copyright © 2011 Pearson Education, Inc.

Truth Table Practice Note: ~ signifies NEGATION  signifies AND  signifies OR Practice by writing the truth values of each row in the table above. Copyright © 2011 Pearson Education, Inc.

Truth Table Practice Note: ~ signifies NEGATION  signifies AND  signifies OR Practice by writing the truth values of each row in the table above. Copyright © 2011 Pearson Education, Inc.

Alternative Phrasings of Conditionals The following are common alternative ways of stating if p, then q: p is sufficient for q p will lead to q p implies q q is necessary for p q if p q whenever p Copyright © 2011 Pearson Education, Inc.

Rephrasing Conditional Propositions CN 10a - c Recast each of the following statements in the form if p, then q. Discuss whether the statement is true of false. a. A rise in seas level will devastate Florida b. A red tag on an item is sufficient to mean it’s on sale. c. Being male is necessary for being president of the United States. Copyright © 2011 Pearson Education, Inc.

Variations on the Conditional Converse: Inverse: Contrapositive: If it is raining, then I will bring an umbrella to work. If I bring an umbrella to work, then it must be raining. If it is not raining, then I will not bring an umbrella to work. If I do not bring an umbrella to work, then it must not be raining. If p, then q If q, then p If ~p, then ~q If ~q, then ~p Copyright © 2011 Pearson Education, Inc.

Logical Equivalence Two statements are logically equivalent if they share the same truth values. logically equivalent logically equivalent Copyright © 2011 Pearson Education, Inc.

Logical Equivalence CN 11a-c Consider the statement If a creature is a whale, then it is a mammal. a. Write its converse, inverse, and contrapositive. b. Evaluate the truth of each statement. c. Which statements are logically equivalent? Copyright © 2011 Pearson Education, Inc.

1B Homework 1A Discussion Paragraph (DP) Class Notes (CN) 1-11 P.31 Quick Quiz 1-10 p. 31: 1-12 1 web 103. Web Searches 1 world 104. Logical Or 105. Multiple Negation 106. Conditional News Copyright © 2011 Pearson Education, Inc.