Verifying Arguments MATH 102 Contemporary Math S. Rook

Overview Section 3.4 in the textbook: – Verifying an argument using truth tables – Verifying an argument using valid forms

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?

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

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)

Verifying an Argument Using Valid Forms

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

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.

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.

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.

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.

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)