3.6 Euler Diagrams and Syllogistic Arguments 155 As with symbolic logic, the premises and the conclusion together form an argument. An example of a syllogistic argument is All beagles are dogs. All dogs bark. All beagles bark. ∴ This is an example of a valid argument. Recall from Section 3.5 that an argument is valid when its conclusion necessarily follows from a given set of premises. Recall that an argument in which the conclusion does not necessarily follow from the given premises is said to be an invalid argument or a fallacy . Before we give another example of a syllogism, let’s review the Venn diagrams discussed in Section 2.3 in relationship with Aristotle’s four statements. Profile in Mathematics Leonhard Euler Swiss mathematician Leonhard Euler (pronounced “oiler”; 1707–1783) produced such a large volume of work that it has been estimated that a complete edition of his work would make up over 70 volumes. Although his greatest contribution was made in the field of modern analysis, his work also explored physics, astronomy, trigonometry, and calculus. Euler also had a great influence on the notation and symbolism of mathematics, and it was through his work that the symbols e and π came into common use. His facility for mental calculation and his uncommon memory allowed him to continue his prolific career even after he lost his vision in both eyes. All A’s are B’s No A’s are B’s Some A’s are B’s Some A’s are not B’s U B A U A B U A B U A B If an element is in set A, then it is in set B. If an element is in set A, then it is not in set B. There is at least one element that is in both set A and set B. There is at least one element that is in set A that is not in set B. One method used to determine whether an argument is valid or is a fallacy is by means of an Euler diagram , named after Leonhard Euler, who used circles to represent sets in syllogistic arguments. The technique of using Euler diagrams is illustrated in Example 1. B K U Figure 3.3 U V B K Figure 3.4 Example 1 Using an Euler Diagram Determine whether the following syllogism is valid or invalid. All keys are made of brass. All things made of brass are valuable. All keys are valuable. ∴ Solution To determine whether this syllogism is valid or invalid, we will construct an Euler diagram. We begin with the first premise, “All keys are made of brass.” As shown in Fig. 3.3, the inner blue circle labeled K represents the set of all keys and the outer red circle labeled B represents the set of all brass objects. The first premise requires that the inner blue circle must be entirely contained within the outer red circle. Next, we will represent the second premise, “All things made of brass are valuable.” As shown in Fig. 3.4, the outermost black circle labeled V represents the set of all valuable objects. The second premise dictates that the red circle, representing the set of brass objects, must be entirely contained within the black circle, representing the set of valuable objects. Now, examine the completed Euler diagram in Fig. 3.4. Note that the premises force the set of keys to be within the set of valuable objects. Therefore, the argument is valid, since the conclusion, “All keys are valuable,” necessarily follows from the set of premises. 7 Now try Exercise 7 The syllogism in Example 1 is valid even though the conclusion, “All keys are valuable,” is not a true statement. Similarly, a syllogism can be invalid, or a fallacy, even if the conclusion is a true statement.
RkJQdWJsaXNoZXIy NjM5ODQ=