146 CHAPTER 3 Logic Once we have demonstrated that an argument in a particular form is valid, all arguments with exactly the same form will also be valid. In fact, many of these forms have been assigned names. The argument form just discussed, p q p q → ∴ is called the law of detachment, or modus ponens. Example 1 Determining the Validity of an Argument Without 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 in the course. You score 90% on the final exam. You will get an A in the course. ∴ Solution Translate the argument into symbolic form. Let p q : You score 90% on the final exam. : You will get an A in the course. In symbolic form, the argument is p q p q → ∴ This argument is the law of detachment. Therefore it is a valid argument. 7 Now try Exercise 35 Next, we summarize the procedure for determining whether an argument is valid or not valid. TO DETERMINE WHETHER AN ARGUMENT IS VALID 1. Write the argument in symbolic form. 2. Compare the form of the argument with forms that are known to be valid or invalid. If there are no known forms to compare the argument to, or you do not remember the forms, go to Step 3. 3. If the argument contains two premises, write a conditional statement of the form [(premise 1) (premise 2)] conclusion ∧ → 4. Construct a truth table for the statement in Step 3. 5. If the answer column of the truth table has all trues, the statement is a tautology, and the argument is valid. If the answer column does not have all trues, the argument is invalid. PROCEDURE Examples 1 through 4 contain two premises. When an argument contains more than two premises, Step 3 of the procedure will change slightly, as will be explained shortly. Example 2 Determining the Validity of an Argument with a Truth Table Determine whether the following argument is valid or invalid. If Detroit is in Michigan, then Dallas is in California. Dallas is not in California. Detroit is not in Michigan. ∴ LStockStudio/Shutterstock
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