Module: Venn diagrams
Quote of the page
What is the hardest task in the world? To think.
- Ralph Waldo Emerson
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:
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.
|Step 1 : We use the W circle to represent the class of whales, the M circle to represent the class of mammals, and the B circle to represent the class of warm-blooded animals.
|Step 2a : We now represent the information in the first premise.
|Step 2b : We now represent the information in the second premise.
Step 3 : We now draw an outline for the conclusion. This is the green 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, the argument is valid.
In order to be more familiar with the method, let us look at a few more syllogisms.
Every A is B.
Some B is C.
Therefore, some A is C.
In the diagram above, we have already drawn a Venn diagram for the three classes and encoded the information contained 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?[Add outline] [Remove outline]
If the argument is valid, there should be a complete tick inside the outlined region. But there is none. So this tells us that the argument is not valid.
Some A is B.
Every B is C.
Therefore, some A is C.
Again, we have already encoded the information in the first two premises. We now need an outline for the conclusion.[Add outline] [Remove outline]
If the argument is valid, there should be a complete tick inside the outlined region. And indeed there is. Although part of the tick is outside the green outlined region, that part of the tick appears only in a shaded area. So in effect we have a complete tick within the green outlined region. This shows that the argument is valid.