3.5 Symbolic Arguments 149 Now we consider an argument that has more than two premises. When an argument contains more than two premises, the statement we test, using a truth table, is formed by taking the conjunction of all the premises as the antecedent of a conditional statement and the conclusion as the consequent of the conditional statement. One example is an argument of the following form p p p c 1 2 3 ∴ To determine whether this argument is valid, we determine the truth table for p p p c [ ] . 1 2 3 ∧ ∧ → When we evaluate p p p [ ], 1 2 3 ∧ ∧ it makes no difference whether we evaluate p p p [( ) ] 1 2 3 ∧ ∧ or p p p [ ( )] 1 2 3 ∧ ∧ because both give the same answer. In Example 6, we evaluate p p p [ ] 1 2 3 ∧ ∧ from left to right; that is, p p p [( ) ]. 1 2 3 ∧ ∧ Example 5 Identifying Common Fallacies in Arguments Determine whether the following arguments are valid or invalid. a) If it is snowing, then we put salt on the driveway. We put salt on the driveway. It is snowing. ∴ b) If it is snowing, then we put salt on the driveway. It is not snowing. We do not put salt on the driveway. ∴ Solution a) Let p q : It is snowing. : We put salt on the driveway. In symbolic form, the argument is written as follows. p q q p → ∴ This argument is in the form of the fallacy of the converse. Therefore, the argument is a fallacy, or invalid. b) Using the same symbols defined in the solution to part (a), in symbolic form, the argument is written as follows. p q p q ~ ~ → ∴ This argument is in the form of the fallacy of the inverse. Therefore, the argument is a fallacy, or invalid. 7 Now try Exercise 39 Timely Tip If you are not sure whether an argument with two premises is one of the standard forms, you can always determine whether a given argument is valid or invalid by using a truth table. To do so, follow the boxed procedure on page 146. In Example 5(b), if you did not recognize that this argument was of the same form as the fallacy of the inverse you could construct the truth table for the conditional statement p q p q [( ) ~ ] ~ → ∧ → The true–false values under the conditional column, ,→ would be T, T, F, T. Since the statement is not a tautology, the argument is invalid. Example 6 An Argument with Three Premises Use a truth table to determine whether the following argument is valid or invalid. If my cell phone company is Verizon, then I can call you free of charge. I can call you free of charge or I can send you a text message. I can send you a text message or my cell phone company is Verizon. My cell phone company is Verizon. ∴
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