1. Section 3.6 Euler Diagrams and Syllogistic Arguments

2. What You Will Learn
Euler diagrams
Syllogistic arguments

3. Syllogistic Arguments
Another form of argument is called a syllogistic argument, better known as syllogism.
The validity of a syllogistic argument is determined by using Euler (pronounced “oiler”) diagrams.

4. Euler Diagrams
One method used to determine whether an argument is valid or is a fallacy.
Uses circles to represent sets in syllogistic arguments.

5. Symbolic Arguments Versus Syllogistic Arguments
Euler diagrams
all are, some are, none are, some are not
Syllogistic argument
Truth tables or by comparison with standard forms of arguments
and, or, not, if-then, if and only if
Symbolic argument
Methods of determining validity
Words or phrases used

6. Example 3: Ballerinas and Athletes
Determine whether the following syllogism is valid or invalid.
All ballerinas are athletic.
Keyshawn is athletic.
∴ Keyshawn is a ballerina.

7. Example 3: Ballerinas and Athletes
Solution
Keyshawn is athletic, so must be placed in the set of athletic people, which is A. We have a choice, as shown above.
All ballerinas, B, are athletic, A.
The conclusion does not necessarily follow from the set of premises. The argument is invalid.

8. Example 4: Parrots and Chickens
Determine whether the following syllogism is valid or invalid.
No parrots eat chicken.
Fletch does not eat chicken.
∴ Fletch is a parrot.

9. Example 4: Parrots and Chickens
Solution
The first premise tells us that parrots and things that eat chicken are disjoint sets—that is, sets that do not intersect.
Fletch is not a parrot, the argument is invalid, or is a fallacy.

10. Example 5: A Syllogism Involving the Word Some
Determine whether the following syllogism is valid or invalid.
All As are Bs.
Some Bs are Cs.
∴ Some As are Cs.

11. Example 5: A Syllogism Involving the Word Some
Solution
The premise “All As are Bs” is illustrated.

12. Example 5: A Syllogism Involving the Word Some
Solution
The premise “Some Bs are Cs” means that there is at least one B that is a C.

13. Example 5: A Syllogism Involving the Word Some
Solution
The first illustrations shows that the conclusion “Some As are Cs,” does not follow, the argument is invalid.

14. Example 6: Fish and Cows
Determine whether the following syllogism is valid or invalid.
No fish are mammals.
All cows are mammals.
∴ No fish are cows.

15. Example 6: Fish and Cows Solution
The first premise tells us that fish and mammals are disjoint sets. The second tells us that the set of cows is a subset of the set of mammals.
The conclusion necessarily follows from the premises and the argument is valid.