Verifying Arguments

1. Verifying Arguments MATH 102 Contemporary Math S. Rook

2. Overview • Section 3.4 in the textbook: • Verifying an argument using truth tables • Verifying an argument using valid forms

3. Verifying Arguments Using Truth Tables

4. Verifying Arguments Using Truth Tables • An argument is a logical exercise comprised of a group of statements called premises and a single statement called a conclusion • i.e. The conclusion is a result of the premises • Consider the argument: If news on inflation is good, then stock prices will increase. News on inflation is good. Therefore, stock prices will increase. • What statements are the premises? • What statement is the conclusion?

5. Verifying Arguments Using Truth Tables (Continued) • An argument is valid when using a truth table if: • Whenever all of its premises are true, the conclusion is true OR • When the conditional formed by anding the premises for the hypothesis and the conclusion is a tautology • A tautology is a statement where every row in a truth table is true • A fallacy is a statement where every row in a truth table is false • e.g. Show whether the argument on the previous slide is valid by using a truth table

6. Verifying Arguments Using Truth Tables (Example) Ex 1: Determine whether the argument is valid by using a truth table. a) If you pay your tuition late, then you will pay a late penalty. You do not pay your tuition late. Therefore, you will not pay a late penalty. b)

7. Verifying an Argument Using Valid Forms

8. Argument Forms • An important fact to realize is that arguments are valid for their structure, not their content • We can also tell whether an argument is valid by learning to recognize valid argument forms • What follows on the next few slides are examples of the most common valid and invalid argument forms

9. Valid Argument Forms • Law of Detachment: If p, then q. p. Therefore, q. • e.g. If the door is locked, then I cannot enter. The door is locked. Therefore, I cannot enter. • Law of Contraposition: If p, then q. Not q. Therefore, not p. • e.g. If I tag the player, then he is out. The player is not out. Therefore, I did not tag the player.

10. Valid Argument Forms (Continued) • Law of Syllogism: If p, then q. If q, then r. Therefore, if p, then r. • e.g. If the store has a sale, I will go shopping. If I go shopping, I will take the car. Therefore, if the store has a sale, I will take the car. • Disjunctive Syllogism:p or q. Not p. Therefore, q. • e.g. We can go to California or we can go to Pennsylvania. We do not go to California. Therefore, we go to Pennsylvania.

11. Invalid Argument Forms • Fallacy of the Converse: If p, then q. q. Therefore, p. • e.g. If I returned the book, then the library is open. The library is open. Therefore, I returned the book. • Fallacy of the Inverse: If p, then q. Not p. Therefore, not q. • e.g. If today is Halloween, then I will get candy. Today is not Halloween. Therefore, I do not get candy.

12. Verifying an Argument Using Valid Forms (Example) Ex 2: Determine whether the argument is valid by identifying its form: a) Either my MP3 player is defective or this download is corrupted. My MP3 player is not defective. Therefore, this download is corrupted. b) If I read the want ads, I will find a job. I find a job. Therefore, I read the want ads.

13. Summary • After studying these slides, you should know how to do the following: • State whether an argument is valid by using a truth table • State whether an argument is valid by identifying its form • Additional Practice: • See the list of suggested problems for 3.4 • Next Lesson: • Graphs, Puzzles, and Map Coloring (Section 4.1)