158 CHAPTER 3 Logic Warm Up Exercises In Exercises 1– 6, fill in the blanks with an appropriate word, phrase, or symbol(s). 1. The validity of a syllogistic argument can be determined using a(n) ________ diagram. Euler 2. If an Euler diagram can be drawn only in a way in which the conclusion necessarily follows from the premises, the syllogistic argument is a(n) ________ argument. Valid 3. If an Euler diagram can be drawn in a way in which the conclusion does not necessarily follow from the premises, the syllogistic argument is a(n) ________ argument. Invalid 4. The following Euler diagram represents the statement ________ A’s are B’s. All U B A 5. The following Euler diagram represents the statement ________ A’s are B’s. No U A B 6. The following Euler diagram represents the statement ________ A’s are B’s. Some U A B U F M C Figure 3.11 Now try Exercise 21 Solution The first premise tells us that fish and mammals are disjoint sets, as shown in Fig. 3.11. The second premise tells us that the set of cows is a subset of the set of mammals. Therefore, the circle representing the set of cows must go within the circle representing the set of mammals. Note that the set of fish and the set of cows cannot be made to intersect without violating a premise. Thus, the conclusion “No fish are cows” necessarily follows from the premises and the argument is valid. Note that we did not say that this conclusion is true, only that the argument is valid. 7 Exercises SECTION 3.6 Practice the Skills/Problem Solving In Exercises 7–26, use an Euler diagram to determine whether the syllogism is valid or invalid. 7. All judges are lawyers. Joseph is a judge. Joseph is a lawyer. ∴ Valid 8. All dogs are canines. Boomer is a canine. Boomer is a dog. ∴ Invalid 9. All yo-yos are toys. All Slinkies are toys. All Slinkies are yo-yos. ∴ Invalid 10. All ladybugs are beetles. All beetles are insects. All ladybugs are insects. ∴ Valid 11. No PCs are Macs. All Dells are PCs. No Dells are Macs. ∴ Valid 12. All American Beauties are roses. No roses are daffodils. No American Beauties are daffodils. ∴ Valid Christian Musat/ Shutterstock
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