3.5 Symbolic Arguments 147 The argument form in Example 2 is an example of the law of contraposition, or modus tollens. Note that the argument in Example 2 is valid even though the conclusion, “Detroit is not in Michigan,” is a false statement. It is also possible to have an invalid argument in which the conclusion is a true statement. When an argument is valid, the conclusion necessarily follows from the premises. It is not necessary for the premises or the conclusion to be true statements in the argument. Solution Let p q : Detroit is in Michigan. : Dallas is in California. In symbolic form, the argument is p q q p ~ ~ → ∴ As we have not tested an argument in this form, we will construct a truth table to determine whether the argument is valid or invalid. We write the argument in the form p q q p [( ) ] , → ∧ ∼ →∼ and construct a truth table (Table 3.34). Since the answer, column 5, has all T’s, the argument is valid. Now try Exercise 37 Table 3.34 p q p q [( ) → ∧ q] ∼ → p∼ T T T F F T F T F F F T T F F T T F F T T F F T T T T T 1 3 2 5 4 7 Example 3 Another Symbolic Argument Determine whether the following argument is valid or invalid. It is raining or the sun is out. It is not raining. The sun is out. ∴ Solution Let p q : It is raining : The sun is out. In symbolic form, the argument is p q p q ~ ∨ ∴ As this form is not one we have already discussed, we will construct a truth table. We write the argument in the form p q p q [( ) ] . ∨ ∧ ∼ → Next we construct a truth table, as shown in Table 3.35. The answer to the truth table, column 5, is true in every case. Therefore, the statement is a tautology, and the argument is valid.
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