Chapter 3 Section 5 - Slide 1 Copyright © 2009 Pearson Education, Inc. AND

Copyright © 2009 Pearson Education, Inc. Chapter 3 Section 5 - Slide 2 Chapter 3 Logic

Copyright © 2009 Pearson Education, Inc. Chapter 3 Section 5 - Slide 3 WHAT YOU WILL LEARN Symbolic arguments and standard forms of arguments Euler diagrams and syllogistic arguments

Copyright © 2009 Pearson Education, Inc. Chapter 3 Section 5 - Slide 4 Section 5 Symbolic Arguments

Chapter 3 Section 5 - Slide 5 Copyright © 2009 Pearson Education, Inc. Symbolic Arguments An argument is valid when its conclusion necessarily follows from a given set of premises. An argument is invalid (or a fallacy) when the conclusion does not necessarily follow from the given set of premises.

Chapter 3 Section 5 - Slide 6 Copyright © 2009 Pearson Education, Inc. Law of Detachment Also called modus ponens. The argument form symbolically written: Premise 1: Premise 2: Conclusion: If [ premise 1 and premise 2 ] then conclusion [ (p  q)  p ]  q

Chapter 3 Section 5 - Slide 7 Copyright © 2009 Pearson Education, Inc. Determine Whether an Argument is Valid Write the argument in symbolic form. Compare the form with forms that are known to be either valid or invalid. If the argument contains two premises, write a conditional statement of the form [(premise 1)  (premise 2)]  conclusion Construct a truth table for the statement above. If the answer column of the table has all trues, the statement is a tautology, and the argument is valid. If the answer column of the table does not have all trues, the argument is invalid.

Chapter 3 Section 5 - Slide 8 Copyright © 2009 Pearson Education, Inc. Example: Determining Validity with a Truth Table Determine whether the following argument is valid or invalid. If you score 90% on the final exam, then you will get an A for the course. You will not get an A for the course.   You do not score 90% on the final exam.

Chapter 3 Section 5 - Slide 9 Copyright © 2009 Pearson Education, Inc. Example: Determining Validity with a Truth Table (continued) Construct a truth table. In symbolic form the argument is: Solution: Let p: You score 90% on the final exam. q: You will get an A in the course. p  q~q~pp  q~q~p

Chapter 3 Section 5 - Slide 10 Copyright © 2009 Pearson Education, Inc. Example: Determining Validity with a Truth Table (continued) pq[(p  q)  ~ q]  ~p~p TTFFTTFF TFTFTFTF F F T T T F T T F F F T F T F T Fill-in the table in order, as follows: Since column 5 has all T’s, the argument is valid T T T T 5

Chapter 3 Section 5 - Slide 11 Copyright © 2009 Pearson Education, Inc. Valid Arguments Law of Detachment Law of Syllogism Law of Contraposition Disjunctive Syllogism

Chapter 3 Section 5 - Slide 12 Copyright © 2009 Pearson Education, Inc. Invalid Arguments Fallacy of the Converse Fallacy of the Inverse

Chapter 3 Section 5 - Slide 13 Copyright © 2009 Pearson Education, Inc. Translate the following argument into symbolic form. Determine whether the argument is valid or invalid. If Jenny gets some rest, then she will feel better. If Jenny feels better, then she will help me paint my bedroom. Therefore, if my bedroom is painted, then Jenny must have gotten some rest.

Chapter 3 Section 5 - Slide 14 Copyright © 2009 Pearson Education, Inc. Translate the following argument into symbolic form. Determine whether the argument is valid or invalid. If Jenny gets some rest, then she will feel better. If Jenny feels better, then she will help me paint my bedroom. Therefore, if my bedroom is painted, then Jenny must have gotten some rest.

Copyright © 2009 Pearson Education, Inc. Chapter 3 Section 5 - Slide 15 Section 6 Euler Diagrams and Syllogistic Arguments

Chapter 3 Section 5 - Slide 16 Copyright © 2009 Pearson Education, Inc. Syllogistic Arguments Another form of argument is called a syllogistic argument, better known as syllogism. The validity of a syllogistic argument is determined by using Euler (pronounced “oiler”) diagrams.

Chapter 3 Section 5 - Slide 17 Copyright © 2009 Pearson Education, Inc. Euler Diagrams One method used to determine whether an argument is valid or is a fallacy. Uses circles to represent sets in syllogistic arguments.

Chapter 3 Section 5 - Slide 18 Copyright © 2009 Pearson Education, Inc. Symbolic Arguments Versus Syllogistic Arguments Euler diagramsall are, some are, none are, some are not Syllogistic argument Truth tables or by comparison with standard forms of arguments and, or, not, if-then, if and only if Symbolic argument Methods of determining validity Words or phrases used

Chapter 3 Section 5 - Slide 19 Copyright © 2009 Pearson Education, Inc. Example: Ballerinas and Athletes Determine whether the following syllogism is valid or invalid. All ballerinas are athletic. Keyshawn is athletic.  Keyshawn is a ballerina.

Chapter 3 Section 5 - Slide 20 Copyright © 2009 Pearson Education, Inc. Example: Ballerinas and Athletes Keyshawn is athletic, so must be placed in the set of athletic people, which is A. We have a choice, as shown above. All ballerinas, B, are athletic, A. A B U The conclusion does not necessarily follow from the set of premises. The argument is invalid. A B U A B U Let A = all Athletes and B = all Ballerinas. Let K represent Keyshawn.

Chapter 3 Section 5 - Slide 21 Copyright © 2009 Pearson Education, Inc. Determine whether the syllogism is valid or is a fallacy. a.Valid b.Fallacy c.Can’t determine Some teachers teach math. Some teachers teach English. Therefore, some teachers teach math and English.

Chapter 3 Section 5 - Slide 22 Copyright © 2009 Pearson Education, Inc. Determine whether the syllogism is valid or is a fallacy. a.Valid b.Fallacy c.Can’t determine Some teachers teach math. Some teachers teach English. Therefore, some teachers teach math and English.