13.6 Euler Diagrams and Syllogistic Arguments


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1 EulerDiagrams.nb Euler Diagrams and Syllogistic rguments In the preceding section, we showed how to determine the validity of symbolic arguments using truth tables and comparing the arguments to standard forms. This section presents another form of argument called a syllogistic argument, better known by the shorter name of syllogism. The validity of a syllogistic argument is determined by using Euler diagrams. Syllogistic logic, a deductive process of arriving at a conclusion, was developed by risotle in about ristotle considered the relationships among the four types of statements that follow. ll are. No are. Some are. Some are not. Examples of these statements are : ll teachers are rich. No teachers are rich. Some teachers are rich. Some teachers are not rich. Since ristotle's time, other types of statements have been added to the study of syllogistic logic, two of which are is a. is not a. Examples of these statements are: oach Zab is a carpenter. oach Zab is not a carpenter. The difference between a symbolic argument and a syllogistic argument can be seen in the following chart. Symbolic arguments use the connectives and, or, not, ifthen, and if and only if. Syllogistic arguments use the quantifiers all, some, and none which we discussed earlier.
2 EulerDiagrams.nb 2 à Symbolic rguments Versus Syllogistic rguments Words or phrases used Method of determining validity Symbolic and, or, not, ifthen, Truth tables or by comparison with argument if and only if standard forms of arguments Syllogistic all are, some are, none Euler diagrams argument are, some are not s with symbolic logic, the premises and the conclusion together form an argument. n example of a syllogistic argument is ll German shepherds are dogs. ll dogs bark. \ ll German shepherds bark. This is an example of a valid argument. Recall from the previous section 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. efore we give another example of a syllogism, let's review the Venn diagrams discussed earlier in relationship with ristotle's four statements. ll 's are 's
3 EulerDiagrams.nb 3 No 's are 's Some 's are 's Some 's are not 's One method used to determine whether an argument is valid or is a fallacy is by means of an Euler diagram, named after Leonard Euler ( ) who used circles to represent sets in syllogistic arguments. The technique of using Euler diagrams is illustrated in the next example.
4 EulerDiagrams.nb 4 Example 1: sing an Euler Diagram Determine whether the following syllogism is valid or is a fallacy. ll horses are brown. ll brown animals have fur. \ ll horses have fur. Solution: The statement "ll horses are brown" may be interpreted as "If an animal is a horse, then it is brown," onstruct the diagram and represent the first premise "ll horse are brown.," as shown below. H The outer circle represents all brown animals, and the inner circle represents all horses. Now illustrate the second premise "ll brown animals have fur," as shown in the next figure. F H The set containing all animals that have fur must contain all the brown animals, as illustrated in the diagram. Note that the premises force the set of horses to be within the set of animals that have fur. Therefore, the argumet is valid, since the conclusion "ll horses have fur" necessarily follows from the premises. The argument in Example 1 is valid even though the conclusion "ll horse have fur" is obvi
5 EulerDiagrams.nb 5 ously a false statement. Similarly, an argument can be a fallacy even if the conclusion is a true statement. 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 ll earth people have two heads. ll people with two heads can fly. \ ll earth people can fly. The form of this argument is the same as that of the previous valid argument. Therefore, this argument is valid. Example 2: Is the Syllogism Valid? Determine whether the following syllogism is valid or is a fallacy. ll pilots have good vision. Kaitlyn is a pilot. \ Kaitlyn has good vision. Solution: The statement "ll pilots have good vision" is illustrated in the figure below. G P The second premise, "Kaitlyn is a pilot", tells us that Kaitlyn must be placed in the inner circle (see below).
6 EulerDiagrams.nb 6 G P àk The Euler diagram illustrates that we must accept the conclusion "Kaitlyn has good vision" as true (when we accept the premises as true). Therefore, the argument is valid. 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 brown animals had to be placed inside the set of animals with fur. In Example 2, Kaitlyn had to be placed inside the set of people with good vision. Often when determining the truth value of a syllogisim, 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 this 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: Is hristine a Football Player? Determine whether the following syllogism is valid or is a fallacy. ll football players are strong. hristine is strong. \ hristine is a football player. Solution: The statement "ll football players are strong" is illustrated in the figure below.
7 EulerDiagrams.nb 7 S F The next premise, "hristine is strong," tells us that hristine must be placed in the set of strong people. Two diagrams in which both premises are satisfied are shown below. S F à S F à y examining the first figure above, however, we see that hristine is not a football player. Therefore, the conclusion "hristine is a footbal player" does not necessariliy follow from the premises. Thus, the argument is not valid.
8 EulerDiagrams.nb 8 Example 4: Parrots and hickens Determine whether the following syllogism is valid or is a fallacy. No parrots eat chicken. Fletch does not eat chicken. \ Fletch is a parrot. Solution: The diagram below satisfies the two given premises and also shows that Fletch is not a parrot. Therefore, the argument is invalid, or is a fallacy. P à F Example 5: Syllogism Involving the Word Some Determine whether the following syllogism is valid or invalid. ll s are s. Some s are s. \ Some s are s. Solution: The statement "ll s are s" is illustrated below.
9 EulerDiagrams.nb 9 The statement "Some s are s" means that there is at least one that is a. We can illustrate this premise in four ways as illustrated below. In all four illustrations, we see that (1) all s are s and (2) some s are s. The conclusion is "Some s are s." Since at least one of the illustrations (the first one) shows that the conclusion does not necessarily follow from the given premises, the argument is invalid. Remember that we are trying to draw the diagram in such a way that the premises are true and the conclusion is false. If this is not possible, then the argument is valid.
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