156 CHAPTER 3 Logic When we determine the validity of an argument, we are determining whether the conclusion necessarily follows from the premises. When we say that an argument is valid, we are saying that if all the premises are true statements, then the conclusion must also be a true statement. The form of the argument determines its validity, not the particular statements. For example, consider the syllogism ∴ All Earth people have two heads. All people with two heads can fly. All Earth people can fly. The form of this argument is the same as that of the previous valid argument in Example 1. Therefore, this argument is also valid. Example 2 Analyzing a Syllogism Determine whether the following syllogism is valid or invalid. All pediatricians are doctors. Teisha is a pediatrician. Teisha is a doctor. ∴ Solution The statement “All pediatricians are doctors” is illustrated in Fig. 3.5. Note that the P circle must be completely inside the D circle. The second premise “Teisha is a pediatrician” tells us that Teisha must be placed in the inner circle labeled P (Fig. 3.6). The Euler diagram illustrates that by placing Teisha inside the P circle, Teisha must also be inside the D circle. Therefore, the conclusion “Teisha is a doctor” necessarily follows from the premises and the argument is valid. 7 Now try Exercise 11 U D P Figure 3.5 P D U T Figure 3.6 In both Example 1 and Example 2, we had no choice as to where the second premise was to be placed in the Euler diagram. In Example 1, the set of brass objects had to be placed inside the set of valuable objects. In Example 2, Teisha had to be placed inside the set of pediatricians. Often when determining the truth value of a syllogism, a premise can be placed in more than one area in the diagram. We always try to draw the Euler diagram so that the conclusion does not necessarily follow from the premises. If that can be done, then the conclusion does not necessarily follow from the premises and the argument is invalid. If we cannot show that the argument is invalid, only then do we accept the argument as valid. We illustrate this process in Example 3. Example 3 Harmonicas and Trumpets Determine whether the following syllogism is valid or invalid. All harmonicas are musical instruments. A trumpet is a musical instrument. A trumpet is a harmonica. ∴ Solution The premise “All harmonicas are musical instruments” is illustrated in Fig. 3.7(a). The next premise, “A trumpet is a musical instrument,” tells us that a trumpet must be placed in the set of musical instruments. Two diagrams in which both premises are satisfied are shown in Fig. 3.7(b) and (c). By examining Fig. 3.7(b), however, we see that a trumpet is not a harmonica. Therefore, the conclusion, “A trumpet is a harmonica,” does not necessarily follow from the set of premises. The argument is thus invalid, or the argument is a fallacy. Learning Catalytics Keyword: Angel-SOM-3.6 (See Preface for additional details.)
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