PART FOUR: VENN DIAGRAMS

Reading information from a Venn diagram

If a Venn diagram completely contains an “X” within a given region, we will understand the diagram as ‘saying’ that there exists an object of the type represented by the region.

If a region is completely shaded, we will understand the diagram as ‘saying’ that there does not exist any objects of the type represented by the region.

If a region is blank (i.e., it is not shaded and does not contain an “X”), then the diagram does not tell us whether there is an object of the kind represented by the region.

Exercise 4.2: After each statement either (i) write “True” to indicate that the Venn diagram shows the statement to be true, (ii) write “False” to indicate that the Venn diagram shows the statement to be false, or (iii) write “Undetermined” to indicate that the Venn diagram does not give enough information to show the statement to be true and does not give enough information to show the statement to be false. The answers for (1), (2), and (3) are given.

Venn diagram test for consistency of a set of sentences:

1. Construct a single Venn diagram for all the sentences in the set.

2. Check to see whether there is an “X” in a completely shaded region.

If there is, the set of sentences is inconsistent.

If there is not, the set of sentences is consistent.

Exercise 4.3: Test each of the following sets of sentences for consistency.

1. Some A are B. All A are B. Some B is not A. No C are A.	2. All A are B. All B are C. All C are A.	3. No A are B. No B are C. No C are A. Some A is not B. Some B is not C. Some C is not A.

4. No A are B. Some C is A. All C are B.	5. No C are B.     All C are A.     Some A is not C.	6. Not every A is B. Not all B are C. Not every A is C.	7. Not all dogs are pets. Not every pet is a dog. Not any dog is a cat. Dog: A / Pets: B / Cats: C

Venn diagram test for validity of an argument:

1. Diagram all the premises (but not the conclusion).

2. Check to see whether an “X” falls completely within a shaded region. If it does, then the premises are inconsistent and thus the argument is valid.

3. Check to see whether the conclusion is shown to be true by the diagram you have just constructed. If it is, the argument is valid.

4. If the argument is not shown to be valid by step (2) or by step (3), then it is invalid.

Note: step (2) is included to cover the rare case in which the premises are inconsistent. Such arguments are treated as valid but unsound.

Exercise 4.4: Test the following arguments for validity using Venn diagrams.

Exercise 4.5: Venn diagrams for simple and complex statements. Represent the following statements using Venn diagrams. (Abbreviations: B: Birds; F: things that fly; S: things that swim).

1. Only birds fly.	2. All and only birds fly.	3. Not all birds fly.

4. Some bird that flies swims.	5. Some bird that flies doesn’t swim.	6. Every bird that flies is a swimmer.

7. Not every bird that flies is a swimmer.	8. Every swimmer that is a bird flies.	9. Every swimmer is a bird that flies.

10. Some flightless bird swims.	11. Every bird that flies is a nonswimmer.	12. No bird that swims also flies.

13. Every bird both flies and swims.	14. Every bird either flies or swims.	15. Every bird either flies or swims. (Use the exclusive “or”.)

Exercise 4.6: More Venn diagram test for validity. Use Venn diagrams to test the following arguments for validity.

Venn Diagram Practice: This is a practice sheet for those having difficulty with Venn diagrams. Fold the sheet lengthwise so that you cannot see the answers at the right. Try diagramming the statements at the left and then check your work. For problems 1-6, try both diagrams.

PART 4 SUPPLEMENTARY EXERCISES WITH ANSWERS

Supplementary Exercise 1.

After each statement either (i) circle “T” to indicate that the Venn diagram shows the statement to be true, (ii) circle “F” to indicate that the Venn diagram shows the statement to be false, or (iii) circle “U” (for “undetermined”) to indicate that the Venn diagram does not give enough information to show the statement to be true and does not give enough information to show the statement to be false.

(1) Some person who has ridden a caribou has been to the North Pole. T F U

(2) Not everyone who has ridden an elephant has been to the North Pole. T F U

(3) Everyone who has ridden a caribou has been to the North Pole. T F U

(4) Everyone who has ridden a caribou has ridden an elephant. T F U

(5) No one who has ridden an elephant has ridden a caribou. T F U

(6) Someone who has not been to the North Pole has ridden an elephant. T F U

(7) Everyone who has ridden a caribou but not an elephant has been to the North Pole. T F U

(8) No one who has ridden an elephant and a caribou has been to the North Pole. T F U

(9) Everyone who has been to the North Pole has ridden either an elephant or a caribou.

(10) No one who has ridden an elephant has been to the North Pole.

(11) Not every person who has ridden an elephant has been to the North Pole.

(12) Not any person who has ridden an elephant has been to the North Pole.

Supplementary Exercise 2: Fill in the Venn diagram to represent the sentence.

(1) No pets are lazy.	(2) Some cats are not lazy.	(3) All pets are lazy.	(4) Some cats are lazy.	(5) Only cats are lazy.

(6) Some cats are lazy pets.	(7) Some cat that is not lazy is a pet.	(8) Every lazy pet is a cat.	(9) Not every lazy cat is a pet.	(10) Some lazy cat is a pet.