Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 1, Unit 1B, Slide 1 Thinking Critically 1

Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 1, Unit 1B, Slide 2 Unit 1B Propositions and Truth Values

Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 1, Unit 1B, Slide 3 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 © 2015, 2011, 2008 Pearson Education, Inc. Chapter 1, Unit 1B, Slide 4 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 © 2015, 2011, 2008 Pearson Education, Inc. Chapter 1, Unit 1B, Slide 5 Example Find the negation of the proposition Amanda is the fastest runner on the team. Write its negation. If the negation is false, is Amanda really the fastest runner on the team? Solution The negation of the given proposition is Amanda is not the fastest runner on the team. If the negation is false, the original statement must be true, meaning that Amanda is the fastest runner on the team.

Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 1, Unit 1B, Slide 6 Double Negation The double negation of a proposition p, not not p, has the same truth value as p.

Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 1, Unit 1B, Slide 7 Example 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 University of North Carolina (Chapel Hill) was asked about the possibility of a similar association among younger workers at another national laboratory. He was quoted as saying (Boulder Daily Camera): My opinion is that it’s unlikely that there is no association. Does the scientist think there is an association between low- level radiation and cancer among younger workers?

Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 1, Unit 1B, Slide 8 Example (cont) Solution Because of the words “unlikely” and “no association,” the scientist’s statement contains a double negation. p = it’s likely that there is an association (between low-level radiation and cancer) The word “unlikely” gives us the statement it’s unlikely that there is an association, which we identify as not p. The words “no association” transform this last statement into the original statement, it’s unlikely that there is no association, which we recognize as not not p.

Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 1, Unit 1B, Slide 9 Example (cont) Because the double negation has the same truth value as the original proposition, we conclude that the scientist believes it likely that there is an association between low-level radiation and cancer among younger workers.

Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 1, Unit 1B, Slide 10 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 Connectors 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 © 2015, 2011, 2008 Pearson Education, Inc. Chapter 1, Unit 1B, Slide 11 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: And Statements (Conjunctions)

Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 1, Unit 1B, Slide 12 Example 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. Solution a. The statement contains two distinct propositions: The capital of France is Paris and Antarctica is cold. Because both propositions are true, their conjunction is also true.

Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 1, Unit 1B, Slide 13 Example 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. b. The statement contains two distinct propositions: The capital of France is Paris and the capital of America is Madrid. Although the first proposition is true, the second is false. Therefore, their conjunction is false.

Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 1, Unit 1B, Slide 14 An inclusive or means “either or both.” An exclusive or means “one or the other, but not both.” The word or can be interpreted in two distinct ways: In logic, assume or is inclusive unless told otherwise. Two Types of Or

Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 1, Unit 1B, Slide 15 The Logic of Or (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 © 2015, 2011, 2008 Pearson Education, Inc. Chapter 1, Unit 1B, Slide 16 Example Consider the statement airplanes can fly or cows can read. Is it true? Solution The statement is a disjunction of two propositions: (1) airplanes can fly; (2) cows can read. The first proposition is clearly true, while the second is clearly false, which makes the disjunction p or q true. That is, the statement airplanes can fly or cows can read is true.

Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 1, Unit 1B, Slide 17 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. The Logic of If... Then Statements (Conditionals)

Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 1, Unit 1B, Slide 18 p is sufficient for q p will lead to q p implies q The following are common alternative ways of stating if p, then q: q is necessary for p q if p q whenever p Alternative Phrasings of Conditionals

Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 1, Unit 1B, Slide 19 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 not p, then not q If not q, then not p Variations on the Conditional

Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 1, Unit 1B, Slide 20 Two statements are logically equivalent if they share the same truth values. Logical Equivalence logically equivalent

Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 1, Unit 1B, Slide 21 Example Consider the true statement if a creature is a whale, then it is a mammal. Write its converse, inverse, and contrapositive. Evaluate the truth of each statement. Which statements are logically equivalent? Solution The statement has the form if p, then q, where p = a creature is a whale and q = a creature is a mammal. Therefore, we find converse (if q, then p): if a creature is a mammal, then it is a whale. This statement is false, because most mammals are not whales.

Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 1, Unit 1B, Slide 22 Example inverse (if not p, then not q): if a creature is not a whale, then it is not a mammal. This statement is also false; for example, dogs are not whales, but they are mammals. contrapositive (if not q, then not p): if a creature is not a mammal, then it is not a whale. Like the original statement, this statement is true, because all whales are mammals.

Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 1, Unit 1B, Slide 23 Example Note that the original proposition and its contrapositive have the same truth value and are logically equivalent. Similarly, the converse and inverse have the same truth value and are logically equivalent.