PHIL102: Introduction to Critical Thinking and Logic

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:

1. The argument has two premises and a conclusion.
2. The argument mentions at most three classes of objects.
3. 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:

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:

1. Draw a Venn diagram with 3 circles.
2. Represent the information in the two premises.
3. 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 ________."
4. See whether the condition that is written down is satisfied. If so, the argument is valid. Otherwise not.

Example #1

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

	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.

Example #2

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 this diagram, 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? If the argument is valid, there should be a complete tick inside the outlined region. But there isn't (there is only part of a tick). So this tells us that the argument is not valid.

Example #3

• 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. Step 4 : 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 outlined region, that part of the tick appears only in a shaded area. So in effect we have a complete tick within the red outlined region. This shows that the argument is valid.

---

Source: Joe Lau and Jonathan Chan, https://philosophy.hku.hk/think/venn/syllogism.php
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License.