148 CHAPTER 3 Logic Table 3.35 p q p q [( ) ∨ ∧ p~ ] → q T T T F F T T T F T F F T F F T T T T T T F F F F T T F 1 3 2 5 4 7 Now try Exercise 41 The argument form in Example 3 is an example of a disjunctive syllogism . Other standard forms of arguments are given in the following chart. MATHEMATICS TODAY Freedom of Speech or Misleading Advertising? The First Amendment to the United States Constitution guarantees that Americans have the freedom of speech. There are, however, limits to what we may say or write. For example, companies are limited in what they can say or write to advertise their products. The Federal Trade Commission (FTC) states that “ads must be truthful and not misleading; that advertisers must have evidence to back up their assertions; and that ads cannot be unfair.” Although most advertisements are truthful and fair, some enter into a “gray area” of truthfulness. Some of these breaches of fairness may be found in logical fallacies either made directly or implicitly by the context of the ad’s wording or artwork. The FTC has the responsibility to take action against companies they believe are violating their policies. For more information regarding actions taken by the FTC against fraudulent advertisers, go to www.ftc.gov. Why This Is Important Although the FTC works to eliminate fraudulent advertising, consumers should be skeptical of some of the claims made by unscrupulous advertisers. The arguments we study in this section can often be applied to the claims made by such advertisers. Standard Forms of Arguments Valid Arguments Law of Detachment p q p q → ∴ Law of Contraposition p q q p ~ ~ → ∴ Law of Syllogism p q q r p r → → ∴ → Disjunctive Syllogism p q p q ~ ∨ ∴ Invalid Arguments Fallacy of the Converse p q q p → ∴ Fallacy of the Inverse p q p q ~ ~ → ∴ As we saw in Example 1, it is not always necessary to construct a truth table to determine whether or not an argument is valid. The next two examples will show how we can identify an argument as one of the standard arguments given in the chart above. Example 4 Identifying a Standard Argument Determine whether the following argument is valid or invalid. If you are on Instagram, then you see my pictures. If you see my pictures, then you know I have a dog. If you are on Instagram, then you know I have a dog. ∴ Solution Let p q r : You are on Instagram. : You see my pictures. : You know I have a dog. In symbolic form, the argument is p q q r p r → → ∴ → The argument is in the form of the law of syllogism. Therefore, the argument is valid, and there is no need to construct a truth table. 7 Now try Exercise 43 Onur ERSIN/Shutterstock
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