** Module: Venn diagrams**

- V00. Introduction
- V01. Basic notation
- V02. Everything and nothing
- V03. Exercises
- V04. Three circles
- V05. Exercises
- V06. Existence 1
- V07. Existence 2
- V08. Syllogism
- V09. Exercises
- V10. Limitations of Venn diagrams

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- Bertrand Russell

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We now see how Venn diagrams can be used to evaluate certain arguments. There are many arguments that cannot be analysed using Venn diagrams. So we shall restrict our attention only to arguments with these properties:

- The argument has two premises and a conclusion.
- The argument mentions at most three classes of objects.
- The premises and the conclusion include only statements of the following form: Every X is Y, Some X is Y, No X is Y.

Here are two examples :

(Premise #1) Every whale is a mammal.

(Premise #2) Every mammal is warm-blooded.

(Conclusion) Every whale is warm-blooded.

(Premise #1) Some fish is sick.

(Premise #2) No chicken is a fish.

(Conclusion) No chicken is sick.

These arguments are sometimes known as *syllogisms*. What we want to determine is whether they are *valid*. In other words, we want to find out whether the conclusions of these arguments follow logically from the premises. To evaluate validity, we want to check whether the conclusion is true in a diagram where the premises are true. Here is the procedure to follow:

- Draw a Venn diagram with 3 circles.
- Represent the information in the two premises.
- Draw an appropriate outline for the conclusion. Fill in the blank in "If the conclusion is true according to the diagram, the outlined region should ________."
- See whether the condition that is written down is satisfied. If so, the argument is valid. Otherwise not.

Let us apply this method to the first argument on this page :

**Step 1** : We use the A circle to represent the class of whales, the B circle to represent the class of mammals, and the C circle to represent the class of warm-blooded animals.

**Step 2a** : We now represent the information in the first premise. (Every whale is a mammal.)

**Step 2b** : We now represent the information in the second premise. (Every mammal is warm-blooded.)

**Step 3** : We now draw an outline for the area that should be shaded to represent the conclusion. (Every whale is warm-blooded.) This is the red outlined region. We write: "If the conclusion is true according to the diagram, the outlined region __should be shaded__."

**Step 4** : Since this is indeed the case, this means that whenever the premises are true, the conclusion must also be true. So the argument is valid.

Let's go through another example:

Every A is B.

Some B is C.

Therefore, some A is C.

We now draw a Venn diagram to represent the two premises:

In the diagram above, we have already drawn a Venn diagram for the three classes and encode the information in the first two premises. To carry out the third step, we need to draw an outline for the conclusion. Do you know where the outline should be drawn? Show outline

Some A is B.

Every B is C.

Therefore, some A is C.

**Step 1** : Representing the first premise.

**Step 2** : Representing the second premise.

**Step 3** : Add outline for conclusion.